Encyclopaedia Britannica, 11th Edition, "Hydromechanics" to "Ichnography" Volume 14, Slice 2
c. But if the weight P was lowered on the ship from a crane on shore,
the vessel would sink bodily a distance P/wA if P was deposited over F; but deposited anywhere else, say over Q on the water-line area, the ship would turn about a line the antipolar of Q with respect to the confocal ellipse, parallel to FF´, at a distance FK from F
FK = (k² - hV/A)/FQ sin QFF´ (2)
through an angle [theta] or a slope of one in m, given by
1 P P V sin [theta] = --- = ----- = --- · -------- FQ sin QFF´, (3) m wA·FK W Ak² - hV
where k denotes the radius of gyration about FF´ of the water-line area. Burning the coal on a voyage has the reverse effect on a steamer.
HYDRODYNAMICS
20. In considering the motion of a fluid we shall suppose it non-viscous, so that whatever the state of motion the stress across any section is normal, and the principle of the normality and thence of the equality of fluid pressure can be employed, as in hydrostatics. The practical problems of fluid motion, which are amenable to mathematical analysis when viscosity is taken into account, are excluded from treatment here, as constituting a separate branch called "hydraulics" (q.v.). Two methods are employed in hydrodynamics, called the Eulerian and Lagrangian, although both are due originally to Leonhard Euler. In the Eulerian method the attention is fixed on a particular point of space, and the change is observed there of pressure, density and velocity, which takes place during the motion; but in the Lagrangian method we follow up a particle of fluid and observe how it changes. The first may be called the statistical method, and the second the historical, according to J. C. Maxwell. The Lagrangian method being employed rarely, we shall confine ourselves to the Eulerian treatment.
_The Eulerian Form of the Equations of Motion._
21. The first equation to be established is the _equation of continuity_, which expresses the fact that the increase of matter within a fixed surface is due to the flow of fluid across the surface into its interior.
In a straight uniform current of fluid of density [rho], flowing with velocity q, the flow in units of mass per second across a plane area A, placed in the current with the normal of the plane making an angle [theta] with the velocity, is [rho]Aq cos [theta], the product of the density [rho], the area A, and q cos [theta] the component velocity normal to the plane.
Generally if S denotes any closed surface, fixed in the fluid, M the mass of the fluid inside it at any time t, and [theta] the angle which the outward-drawn normal makes with the velocity q at that point,
dM/dt = rate of increase of fluid inside the surface, (1)
= flux across the surface into the interior _ _ / / = - | | [rho]q cos [theta] dS, _/_/
the integral equation of continuity.
In the Eulerian notation u, v, w denote the components of the velocity q parallel to the coordinate axes at any point (x, y, z) at the time t; u, v, w are functions of x, y, z, t, the independent variables; and d is used here to denote partial differentiation with respect to any one of these four independent variables, all capable of varying one at a time.
To transfer the integral equation into the differential equation of continuity, Green's transformation is required again, namely, _ _ _ _ _ / / / /d[xi] d[eta] d[zeta] \ / / | | | ( ----- + ------ + ------- )dx dy dz = | | (l[xi] + m[eta] + n[zeta]) dS, (2) _/_/_/ \ dx dy dz / _/_/
or individually _ _ _ _ _ / / / d[xi] / / | | | ----- dx dy dz = | | l[xi] dS,..., (3) _/_/_/ dx _/_/
where the integrations extend throughout the volume and over the surface of a closed space S; l, m, n denoting the direction cosines of the outward-drawn normal at the surface element dS, and [xi], [eta], [zeta] any continuous functions of x, y, z.
The integral equation of continuity (1) may now be written _ _ _ _ _ / / / d[rho] / / | | | ----- dx dy dz = | | (l[rho]u + m[rho]v + n[rho]w) dS = 0, (4) _/_/_/ dt _/_/
which becomes by Green's transformation _ _ _ / / / /d[rho] d([rho]u) d([rho]v) d([rho]w)\ | | | ( ------ + --------- + --------- + -------- ) dx dy dz = 0, (5) _/_/_/ \ dt dx dy dz /
leading to the differential equation of continuity when the integration is removed.
22. The equations of motion can be established in a similar way by considering the rate of increase of momentum in a fixed direction of the fluid inside the surface, and equating it to the momentum generated by the force acting throughout the space S, and by the pressure acting over the surface S.
Taking the fixed direction parallel to the axis of x, the time-rate of increase of momentum, due to the fluid which crosses the surface, is _ _ _ _ / / / / - | | [rho]uq cos [theta] dS = - | | (l[rho]u² + m[rho]uv + n[rho]uw) dS, (1) _/_/ _/_/
which by Green's transformation is _ _ _ / / / /d([rho]u²) d([rho]uv) d([rho]uw)\ - | | | (---------- + ---------- + ---------- ) dx dy dz. (2) _/_/_/ \ dx dy dz /
The rate of generation of momentum in the interior of S by the component of force, X per unit mass, is _ _ _ / / / | | | [rho]X dx dy dz, (3) _/_/_/
and by the pressure at the surface S is _ _ _ _ _ / / / / / dp - | | lp dS = - | | | -- dx dy dz, (4) _/_/ _/_/_/ dx
by Green's transformation.
The time rate of increase of momentum of the fluid inside S is _ _ _ / / / d([rho]u) | | | --------- dx dy dz; (5) _/_/_/ dt
and (5) is the sum of (1), (2), (3), (4), so that _ _ _ / / / /d[rho]u d[rho]u² d[rho]uv d[rho]uw dp \ | | | ( ------- + -------- + -------- + -------- - [rho]X + -- ) dx dy dz = 0, (6) _/_/_/ \ dt dx dy dz dx /
leading to the differential equation of motion
d[rho]u d[rho]u² d[rho]uv, d[rho]uw dp ------- + -------- + -------- + -------- = [rho]X - --, (7) dt dx dy dz dx
with two similar equations.
The absolute unit of force is employed here, and not the gravitation unit of hydrostatics; in a numerical application it is assumed that C.G.S. units are intended.
These equations may be simplified slightly, using the equation of continuity (5) § 21; for
d[rho]u d[rho]u² d[rho]uv d[rho]uw ------- + -------- + -------- + -------- dt dx dy dz
/du du du du \ = [rho] ( -- + u-- + v-- + w-- ) \dt dx dy dz /
/d[rho] d[rho]u d[rho]v d[rho]w \ + u ( ------ + ------- + ------- + ------- ), (8) \ dt dx dy dz /
reducing to the first line, the second line vanishing in consequence of the equation of continuity; and so the equation of motion may be written in the more usual form
du du du du 1 dp -- + u-- + v-- + w-- = X - ----- --, (9) dt dx dy dz [rho] dx
with the two others
dv dv dv dv 1 dp -- + u-- + v-- + w-- = Y - ----- --, (10) dt dx dy dz [rho] dy
dw dw dw dw 1 dp -- + u-- + v-- + w-- = Z - ----- --. (11) dt dx dy dz [rho] dz
23. As a rule these equations are established immediately by determining the component acceleration of the fluid particle which is passing through (x, y, z) at the instant t of time considered, and saying that the reversed acceleration or kinetic reaction, combined with the impressed force per unit of mass and pressure-gradient, will according to d'Alembert's principle form a system in equilibrium.
To determine the component acceleration of a particle, suppose F to denote any function of x, y, z, t, and investigate the time rate of F for a moving particle; denoting the change by DF/dt,
DF F(x + u[delta]t, y + v[delta]t, z + w[delta]t, t + [delta]t) - F(x, y, z, t) -- = lt·---------------------------------------------------------------------------- dt [delta]t
dF dF dF dF = -- + u-- + v-- + w--; (1) dt dx dy dz
and D/dt is called particle differentiation, because it follows the rate of change of a particle as it leaves the point x, y, z; but
dF/dt, dF/dx, dF/dy, dF/dz (2)
represent the rate of change of F at the time t, at the point, x, y, z, fixed in space.
The components of acceleration of a particle of fluid are consequently
Du du du du du -- = -- + u-- + v-- + w--, (3) dt dt dx dy dz
Dv dv dv dv dv -- = -- + u-- + v-- + w--, (4) dt dt dx dy dz
Dw dw dw dw dw -- = -- + u-- + v-- + w--, (5) dt dt dx dy dz
leading to the equations of motion above.
If F (x, y, z, t) = 0 represents the equation of a surface containing always the same particles of fluid,
DF dF dF dF dF -- = 0, or -- + u-- + v-- + w-- = 0, (6) dt dt dx dy dz
which is called the differential equation of the _bounding surface_. A bounding surface is such that there is no flow of fluid across it, as expressed by equation (6). The surface always contains the same fluid inside it, and condition (6) is satisfied over the complete surface, as well as any part of it.
But turbulence in the motion will vitiate the principle that a bounding surface will always consist of the same fluid particles, as we see on the surface of turbulent water.
24. To integrate the equations of motion, suppose the impressed force is due to a potential V, such that the force in any direction is the rate of diminution of V, or its downward gradient; and then
X = -dV/dx, Y = -dV/dy, Z = -dV/dz; (1)
and putting
dw dv du dw dv du -- - -- = 2[xi], -- - -- = 2[eta], -- - -- = 2[zeta], (2) dy dz dz dx dx dy
d[xi] d[eta] d[zeta] ----- + ------ + ------- = 0, (3) dx dy dz
the equations of motion may be written
du dH -- - 2v[zeta] + 2w[eta] + -- = 0, (4) dt dx
dv dH -- - 2w[xi] + 2u[zeta] + -- = 0, (5) dt dy
dw dH -- - 2u[eta] + 2w[xi] + -- = 0, (6) dt dz
where _ / H = | dp/[rho] + V + ½q², (7) _/
q² = u² + v² + w², (8)
and the three terms in H may be called the pressure head, potential head, and head of velocity, when the gravitation unit is employed and ½q² is replaced by ½q²/g.
Eliminating H between (5) and (6)
D[xi] du dv dw / du dv dw \ ----- - [xi]-- - [eta]-- - [zeta]-- + [xi]( -- + -- + -- ) = 0, (9) dt dx dx dx \ dx dy dz /
and combining this with the equation of continuity
1 D[rho] du dv dw ----- ------ + -- + -- + -- = 0, (10) [rho] dt dx dy dz
we have
D /[xi] \ [xi] du [eta] dv [zeta] dw -- ( ----- ) - ----- -- - ----- -- - ------ -- = 0, (11) dt \[rho]/ [rho] dx [rho] dx [rho] dx
with two similar equations.
Putting
[omega]² = [xi]² + [eta]² + [zeta]², (12)
a _vortex line_ is defined to be such that the tangent is in the direction of [omega], the resultant of [xi], [eta], [zeta], called the components of molecular rotation. A small sphere of the fluid, if frozen suddenly, would retain this angular velocity.
If [omega] vanishes throughout the fluid at any instant, equation (11) shows that it will always be zero, and the fluid motion is then called _irrotational_; and a function [phi] exists, called the _velocity function_, such that
udx + vdy + wdz = -d[phi], (13)
and then the velocity in any direction is the space-decrease or downward gradient of [phi].
25. But in the most general case it is possible to have three functions [phi], [psi], m of x, y, z, such that
udx + vdy + wdz = -d[phi] - md[psi], (1)
as A. Clebsch has shown, from purely analytical considerations (_Crelle_, lvi.); and then
d([psi], m) d([psi], m) d([psi], m) [xi] = ½ -----------, [eta] = ½ -----------, [zeta] = ½ -----------, (2) d(y, z) d(z, x) d(x, y)
and
d[psi] d[psi] d[psi] dm dm dm [xi]------ + [eta]------ + [zeta]------ = 0, [xi]-- + [eta]-- + [zeta]-- = 0, (3) dx dy dz dx dy dz
so that, at any instant, the surfaces over which [psi] and m are constant intersect in the vortex lines.
Putting
d[phi] d[psi] H - ------ - m ------ = K, (4) dt dt the equations of motion (4), (5), (6) § 24 can be written
dK d([psi],m) -- - 2u[zeta] + 2w[eta] - ---------- = 0, ..., ...; (5) dx d(x,t)
and therefore
dK dK dK [xi]-- + [eta]-- + [zeta]-- = 0. (6) dx dy dz
Equation (5) becomes, by a rearrangement,
dK d[psi] /dm dm dm dm \ -- - ------ ( -- + u-- + v-- + w-- ) dx dx \dt dx dy dz /
dm / d[psi] d[psi] d[psi] d[psi] \ + -- ( ------ + u------ + v------ + w------ ) = 0, ..., ..., (7) dx \ dt dx dy dz /
dK d[psi] Dm dm D[psi] -- - ------ -- + -- ------ = 0, ..., ..., (8) dx dx dt dx dt
and as we prove subsequently (§ 37) that the vortex lines are composed of the same fluid particles throughout the motion, the surface m and [psi] satisfies the condition of (6) § 23; so that K is uniform throughout the fluid at any instant, and changes with the time only, and so may be replaced by F(t).
26. When the motion is _steady_, that is, when the velocity at any point of space does not change with the time,
dK -- - 2v[zeta] + 2w[eta] = 0, ..., ... (1) dx
dK dK dK dK dK dK [xi]-- + [eta]-- + [zeta]-- = 0, u-- + v-- + w-- = 0, (2) dx dy dz dx dy dz and _ / K = | dp/[rho] + V + ½q² = H (3) _/
is constant along a vortex line, and a _stream line_, the path of a fluid particle, so that the fluid is traversed by a series of H surfaces, each covered by a network of stream lines and vortex lines; and if the motion is irrotational H is a constant throughout the fluid.
Taking the axis of x for an instant in the normal through a point on the surface H = constant, this makes u = 0, [xi] = 0; and in steady motion the equations reduce to
dH/d[nu] = 2v[zeta] - 2w[eta] = 2q[omega] sin [theta], (4)
where [theta] is the angle between the stream line and vortex line; and this holds for their projection on any plane to which d[nu] is drawn perpendicular.
In plane motion (4) reduces to
dH / dQ q \ ----- = 2q[zeta] = q ( -- + --- ), (5) d[nu] \ dv r /
if r denotes the radius of curvature of the stream line, so that
1 dp dV dH d½q² q² ----- ----- + ----- = ----- - ----- = ---, (6) [rho] d[nu] d[nu] d[nu] d[nu] r
the normal acceleration.
The osculating plane of a stream line in steady motion contains the resultant acceleration, the direction ratios of which are
du du du d½q² d½q² dH u-- + v-- + w-- = ---- - 2v[zeta] + 2w[eta] = ---- - --, ..., (7) dx dy dz dx dx dx
and when q is stationary, the acceleration is normal to the surface H = constant, and the stream line is a geodesic.
Calling the sum of the pressure and potential head the statical head, surfaces of constant statical and dynamical head intersect in lines on H, and the three surfaces touch where the velocity is stationary.
Equation (3) is called Bernoulli's equation, and may be interpreted as the balance-sheet of the energy which enters and leaves a given tube of flow.
If homogeneous liquid is drawn off from a vessel so large that the motion at the free surface at a distance may be neglected, then Bernoulli's equation may be written
H = p/[rho] + z + q²/2g = P/[rho] + h, (8)
where P denotes the atmospheric pressure and h the height of the free surface, a fundamental equation in hydraulics; a return has been made here to the gravitation unit of hydrostatics, and Oz is taken vertically upward.
In particular, for a jet issuing into the atmosphere, where p = P,
q²/2g = h - z, (9)
or the velocity of the jet is due to the head k - z of the still free surface above the orifice; this is Torricelli's theorem (1643), the foundation of the science of hydrodynamics.
27. _Uniplanar Motion._--In the uniplanar motion of a homogeneous liquid the equation of continuity reduces to
du dv -- + -- = 0, (1) dx dy
so that we can put
u = -d[psi]/dy, v = d[psi]/dx, (2)
where [psi] is a function of x, y, called the stream- or current-function; interpreted physically, [psi] - [psi]0, the difference of the value of [psi] at a fixed point A and a variable point P is the flow, in ft.³/second, across any curved line AP from A to P, this being the same for all lines in accordance with the continuity.
Thus if d[psi] is the increase of [psi] due to a displacement from P to P´, and k is the component of velocity normal to PP´, the flow across PP´ is d[psi] = k·PP´; and taking PP´ parallel to Ox, d[psi] = vdx; and similarly d[psi]= -udy with PP´ parallel to Oy; and generally d[psi]/ds is the velocity across ds, in a direction turned through a right angle forward, against the clock.
In the equations of uniplanar motion
dv du d²[psi] d²[psi] 2[zeta] = -- - -- = ------ + ------ = -[Nabla]²[psi], suppose, (3) dx dy dx² dy²
so that in steady motion
dH d[psi] dH d[psi] dH -- + [Nabla]²[psi]------ = 0, -- + [Nabla]²[psi]------ = 0, ------ + [Nabla]²[psi] = 0, (4) dx dx dy dy d[psi]
and [Nabla]²[psi] must be a function of [psi].
If the motion ia irrotational,
d[phi] d[psi] d[phi] d[psi] u = - ------ = - ------, v = - ----- = ------, (5) dx dy dy dx´
so that [psi] and [phi] are conjugate functions of x and y,
[phi] + [psi]i = [f](x + yi), [Nabla]²[psi] = 0, [Nabla]²[phi] = 0; (6)
or putting
[phi] + [psi]i = w, x + yi = z, w = [f](z).
The curves [phi] = constant and [psi] = constant form an orthogonal system; and the interchange of [phi] and [psi] will give a new state of uniplanar motion, in which the velocity at every point is turned through a right angle without alteration of magnitude.
For instance, in a uniplanar flow, radially inward towards O, the flow across any circle of radius r being the same and denoted by 2[pi]m, the velocity must be m/r, and
[phi] = m log r, [psi] = m[theta], [phi] + [psi]i = m log re^(i[theta]), w = m log z. (7)
Interchanging these values
[psi] = m log r, [phi] = m[theta], [psi] + [phi]i = m log re^(i[theta]) (8)
gives a state of vortex motion, circulating round Oz, called a straight or columnar vortex.
A single vortex will remain at rest, and cause a velocity at any point inversely as the distance from the axis and perpendicular to its direction; analogous to the magnetic field of a straight electric current.
If other vortices are present, any one may be supposed to move with the velocity due to the others, the resultant stream-function being
[psi] = [Sigma]m log r = log [Pi]r^m; (9)
the path of a vortex is obtained by equating the value of [psi] at the vortex to a constant, omitting the r^m of the vortex itself.
When the liquid is bounded by a cylindrical surface, the motion of a vortex inside may be determined as due to a series of vortex-images, so arranged as to make the flow zero across the boundary.
For a plane boundary the image is the optical reflection of the vortex. For example, a pair of equal opposite vortices, moving on a line parallel to a plane boundary, will have a corresponding pair of images, forming a rectangle of vortices, and the path of a vortex will be the Cotes' spiral
r sin 2[theta] = 2a, or x^(-2) + y^(-2) = a^(-2); (10)
this is therefore the path of a single vortex in a right-angled corner; and generally, if the angle of the corner is [pi]/n, the path is the Cotes' spiral
r sin n[theta] = na. (11)
A single vortex in a circular cylinder of radius a at a distance c from the centre will move with the velocity due to an equal opposite image at a distance a²/c, and so describe a circle with velocity
mc/(a² - c²) in the periodic time 2[pi](a² - c²)/m. (12)
Conjugate functions can be employed also for the motion of liquid in a thin sheet between two concentric spherical surfaces; the components of velocity along the meridian and parallel in colatitude [theta] and longitude [lambda] can be written
d[phi] 1 d[psi] 1 d[psi] d[psi] -------- = ----------- ---------, ----------- --------- = - --------, (13) d[theta] sin [theta] d[lambda] sin [theta] d[lambda] d[theta]
and then
[phi] + [psi]i = F(tan ½[theta]·e^([lambda]i)). (14)
28. _Uniplanar Motion of a Liquid due to the Passage of a Cylinder through it._--A stream-function [psi] must be determined to satisfy the conditions
[Nabla]²[psi] = 0, throughout the liquid; (1)
[psi] = constant, over any fixed boundary; (2)
d[psi]/ds = normal velocity reversed over a solid boundary, (3)
so that, if the solid is moving with velocity U in the direction Ox, d[psi]/ds = -Udy/ds, or [psi] + Uy = constant over the moving cylinder; and [psi] + Uy = [psi]´ is the stream function of the relative motion of the liquid past the cylinder, and similarly [psi] - Vx for the component velocity V along Oy; and generally
[psi]´ = [psi] + Uy - Vx (4)
is the relative stream-function, constant over a solid boundary moving with components U and V of velocity.
If the liquid is stirred up by the rotation R of a cylindrical body,
d[psi]/ds = normal velocity reversed
dx dy = -Rx-- - Ry--, (5) ds ds
[psi] + ½R(x² + y²) = [psi]´, (6)
a constant over the boundary; and [psi]´ is the current-function of the relative motion past the cylinder, but now
V²[psi]´ + 2R = 0, (7)
throughout the liquid.
Inside an equilateral triangle, for instance, of height h,
[psi]´ = -2R[alpha][beta][gamma]/h, (8)
where [alpha], [beta], [gamma] are the perpendiculars on the sides of the triangle.
In the general case [psi]´ = [psi] + Uy - Vx + ½R(x² + y²) is the relative stream function for velocity components, U, V, R.
29. _Example 1._--Liquid motion past a circular cylinder.
Consider the motion given by
[omega] = U(z + a²/z), (1)
so that
/ a²\ / a² \ [phi] = U ( r + -- ) cos [theta] = U ( 1 + -- )x, (2) \ r / \ r² /
/ a²\ / a² \ [psi] = U ( r - -- ) sin [theta] = U ( 1 - -- )y. \ r / \ r² /
Then [psi] = 0 over the cylinder r = a, which may be considered a fixed post; and a stream line past it along which [psi] = Uc, a constant, is the curve
/ a²\ ( r - -- ) sin [theta] = c, (x² + y²)(y - c) - a²y = 0 (3) \ r /
a cubic curve (C3).
Over a concentric cylinder, external or internal, of radius r = b,
/ a²\ [psi]´ = [psi] + U1y = [U ( 1 - --- ) + U1] y, (4) \ b²/
and [psi]´ is zero if
U1/U = (a² - b²)/b²; (5)
so that the cylinder may swim for an instant in the liquid without distortion, with this velocity U1, and [omega] in (1) will give the liquid motion in the interspace between the fixed cylinder r = a and the concentric cylinder r = b, moving with velocity U1.
When b = 0, U1 = [oo]; and when b = [oo], U1 = -U, so that at infinity the liquid is streaming in the direction xO with velocity U.
If the liquid is reduced to rest at infinity by the superposition of an opposite stream given by [omega] = -Uz, we are left with
[omega] = Ua²/z, (6)
[phi] = U(a²/r) cos [theta] = Ua²x/(x² + y²), (7)
[psi] = -U(a²/r) sin [theta] = -Ua²y/(x² + y²), (8)
giving the motion due to the passage of the Cylinder r = a with velocity U through the origin O in the direction Ox.
If the direction of motion makes an angle [theta]´ with Ox,
d[phi] / d[phi] 2xy tan[theta]´ = ----- / ----- = ------ = tan 2[theta], [theta] = ½[theta]´, (9) dy / dx x² - y²
and the velocity is Ua²/r².
Along the path of a particle, defined by the C3 of (3),
y² y(y - c) sin² ½[theta]´ = ------- = -------, (10) x² + y² a²
d[theta]´ 2y - c dy ½ sin [theta]´ --------- = ------ --, (11) ds a² ds
on the radius of curvature is ¼a²/(y - ½c), which shows that the curve is an Elastica or Lintearia. (J. C. Maxwell, _Collected Works_, ii. 208.)
If [phi]1 denotes the velocity function of the liquid filling the cylinder r = b, and moving bodily with it with velocity U1,
[phi]1 = -U1x, (12)
and over the separating surface r = b
[phi] U / a²\ a² + b² --------- = - -- ( 1 + -- ) = -------, (13) [phi]1 U1 \ b²/ a² - b²
and this, by § 36, is also the ratio of the kinetic energy in the annular interspace between the two cylinders to the kinetic energy of the liquid moving bodily inside r = b.
Consequently the inertia to overcome in moving the cylinder r = b, solid or liquid, is its own inertia, increased by the inertia of liquid (a² + b²)/(a² - b²) times the volume of the cylinder r = b; this total inertia is called the effective inertia of the cylinder r = b, at the instant the two cylinders are concentric.
With liquid of density [rho], this gives rise to a kinetic reaction to acceleration dU/dt, given by
a² + b² dU a² + b² dU [pi][rho]b² ------- -- = ------- M´--, (14) a² - b² dt a² - b² dt
if M´ denotes the mass of liquid displaced by unit length of the cylinder r = b. In particular, when a = [oo], the extra inertia is M´.
When the cylinder r = a is moved with velocity U and r = b with velocity U1 along Ox,
a² / b² \ b² / a²\ [phi] = U ------- ( --- + r ) cos [theta] - U1------- ( r + --- ) cos [theta], (15) b² - a² \ r / b² - a² \ r /
a² / b² \ b² / a²\ [psi] = -U ------- ( --- - r ) sin [theta] - U1------- ( r - --- ) sin [theta]; (16) b² - a² \ r / b² - a² \ r /
and similarly, with velocity components V and V1 along Oy
a² / b² \ b² / a²\ [phi] = V ------- ( --- + r ) sin [theta] - V1------- ( r + --- ) sin [theta], (17) b² - a² \ r / b² - a² \ r /
a² / b² \ b² / a²\ [psi] = V ------- ( --- - r ) cos [theta] + V1------- ( r - --- ) cos [theta], (18) b² - a² \ r / b² - a² \ r /
and then for the resultant motion a² z a²b² U + Vi w = (U² + V²) ------- ------ + ------- ------ b² - a² U + Vi b² - a² z
a² z a²b² U1 + V1i -(U1² + V1²) ------- -------- + ------- --------. (19) b² - a² U1 + V1i b² - a² z
The resultant impulse of the liquid on the cylinder is given by the component, over r = a (§ 36), _ / / b² + a² 2b² \ X = | [rho][phi] cos [theta]·ad[theta] = [pi][rho]a² ( U ------- - U1 ------- ); (20) _/ \ b² - a² b² - a²/
and over r = b _ / / 2a² b² + a² \ X1 = | [rho][phi] cos [theta]·bd[theta] = [pi][rho]b² ( U ------ - U1------- ), (21) _/ \ b² - a² b² - a² /
and the difference X - X1 is the component momentum of the liquid in the interspace; with similar expressions for Y and Y1.
Then, if the outside cylinder is free to move
V1 2a² b² - a² X1 = 0, -- = -------, X = [pi][rho]a²U -------. (22) U b² + a² b² + a²
But if the outside cylinder is moved with velocity U1, and the inside cylinder is solid or filled with liquid of density [sigma],
U1 2[rho]b² X = -[pi][rho]a²U, -- = --------------------------------, U [rho](b² + a²) + [sigma](b² - a²)
U - U1 ([rho] - [sigma])(b² - a²) ------ = ---------------------------------, (23) U1 [rho](b² + a²) + [sigma](b² - a²)
and the inside cylinder starts forward or backward with respect to the outside cylinder, according as [rho] > or < [sigma].
30. The expression for [omega] in (1) § 29 may be increased by the addition of the term
im log z = -m[theta] + im log r, (1)
representing vortex motion circulating round the annulus of liquid.
Considered by itself, with the cylinders held fixed, the vortex sets up a circumferential velocity m/r on a radius r, so that the angular momentum of a circular filament of annular cross section dA is [rho]mdA, and of the whole vortex is [rho]m[pi](b² - a²).
Any circular filament can be started from rest by the application of a circumferential impulse [pi][rho]mdr at each end of a diameter; so that a mechanism attached to the cylinders, which can set up a uniform distributed impulse [pi][rho]m across the two parts of a diameter in the liquid, will generate the vortex motion, and react on the cylinder with an impulse couple -[rho]m[pi]a² and [rho]m[pi]b², having resultant [rho]m[pi](b² - a²), and this couple is infinite when b = [oo], as the angular momentum of the vortex is infinite. Round the cylinder r = a held fixed in the U current the liquid streams past with velocity
q´ = 2U sin [theta] + m/a; (2)
and the loss of head due to this increase of velocity from U to q´ is
q´² - U² (2U sin [theta] + m/a)² - U² -------- = ----------------------------, (3) 2g 2g
so that cavitation will take place, unless the head at a great distance exceeds this loss.
The resultant hydrostatic thrust across any diametral plane of the cylinder will be modified, but the only term in the loss of head which exerts a resultant thrust on the whole cylinder is 2mU sin[theta]/ga, and its thrust is 2[pi][rho]mU absolute units in the direction Cy, to be counteracted by a support at the centre C; the liquid is streaming past r = a with velocity U reversed, and the cylinder is surrounded by a vortex. Similarly, the streaming velocity V reversed will give rise to a thrust 2[pi][rho]mV in the direction xC.
Now if the cylinder is released, and the components U and V are reversed so as to become the velocity of the cylinder with respect to space filled with liquid, and at rest at infinity, the cylinder will experience components of force per unit length
(i.) - 2[pi][rho]mV, 2[pi][rho]mU, due to the vortex motion;
(ii.) - [pi][rho]a² dU/dt, -[pi][rho]a² dV/dt, due to the kinetic reaction of the liquid;
(iii.) 0, -[pi]([sigma] - [rho])a²g, due to gravity,
taking Oy vertically upward, and denoting the density of the cylinder by [sigma]; so that the equations of motion are
dU dU [pi][rho]a²-- = - [pi][rho]a²-- - 2[pi][rho]mV, (4) dt dt
dV dV [pi][rho]a²-- = - [pi][rho]a²-- + 2[pi][rho]mV - [pi]([sigma] - [rho])a²g, (5) dt dt
or, putting m = a²[omega], so that the vortex velocity is due to an angular velocity [omega] at a radius a,
([sigma] + [rho])dU/dt + 2[rho][omega]V = 0, (6)
([sigma] + [rho])dV/dt - 2[rho][omega]U + ([sigma] - [rho])g = 0. (7)
Thus with g = 0, the cylinder will describe a circle with angular velocity 2[rho][omega]/([sigma] + [rho]), so that the radius is ([sigma] + [rho])v/2[rho][omega], if the velocity is v. With [sigma] = 0, the angular velocity of the cylinder is 2[omega]; in this way the velocity may be calculated of the propagation of ripples and waves on the surface of a vertical whirlpool in a sink.
Restoring [sigma] will make the path of the cylinder a trochoid; and so the swerve can be explained of the ball in tennis, cricket, baseball, or golf.
Another explanation may be given of the sidelong force, arising from the velocity of liquid past a cylinder, which is encircled by a vortex. Taking two planes x = ± b, and considering the increase of momentum in the liquid between them, due to the entry and exit of liquid momentum, the increase across dy in the direction Oy, due to elements at P and P´ at opposite ends of the diameter PP´, is
[rho]dy (U - Ua²r^(-2) cos 2[theta] + mr^(-1) sin [theta])(Ua²r^(-2) sin 2[theta] + mr^(-1) cos [theta]) + [rho]dy (- U + Ua²r^(-2) cos 2[theta] + mr^(-1) sin [theta])(Ua²r^(-2) sin 2[theta] - mr^(-1) cos [theta]) = 2[rho]dymUr^(-1)(cos [theta] - a^2r^(-2)cos 3[theta]), (8)
and with y = b tan [theta], r = b sec [theta], this is
2[rho]mUd[theta] (1 - a²b^(-2) cos 3[theta] cos [theta]), (9)
and integrating between the limits [theta] = ±½[pi], the resultant, as before, is 2[pi][rho]mU.
31. _Example 2.--Confocal Elliptic Cylinders._--Employ the elliptic coordinates [eta], [xi], and [zeta] = [eta] + [xi]i, such that
z = c ch[zeta], x = c ch [eta] cos [xi], y = c sh [eta] sin [zeta]; (1)
then the curves for which [eta] and [xi] are constant are confocal ellipses and hyperbolas, and
d(x, y) J = -------, [xi]) = c²(ch²[eta] - cos² [xi]) d([eta]
= ½c²(ch 2[eta] - cos 2[xi]) = r1r2 = OD², (2)
if OD is the semi-diameter conjugate to OP, and r1, r2 the focal distances,
r1, r2 = c(ch[eta] ± cos [xi]); (3)
r² = x² + y² = c²(ch²[eta] - sin² [xi])
= ½c²(ch 2[eta] + cos 2[xi]). (4)
Consider the streaming motion given by
w = m ch([zeta] - [gamma]), [gamma] = [alpha] + [beta]i, (5)
[phi] = m ch([eta] - [alpha]) cos ([xi] - [beta]), [psi] = m sh([eta] - [alpha]) sin ([xi] - [beta]). (6)
Then [psi] = 0 over the ellipse [eta] = [alpha], and the hyperbola [xi] = [beta], so that these may be taken as fixed boundaries; and [psi] is a constant on a C4.
Over any ellipse [eta], moving with components U and V of velocity,
[psi]´ = [psi] + Uy - Vx = [m sh([eta] - [alpha]) cos [beta] + Uc sh[eta]] sin [xi]
-[m sh ([eta] - [alpha]) sin [beta] + Vc ch [eta] cos [xi]; (7)
so that [psi]´ = 0, if
m sh([eta] - [alpha]) m sh([eta] - [alpha]) U = - --- ------------------- cos [beta], V = - --- - ------------------- sin [beta], (8) c sh[eta] c ch[eta]
having a resultant in the direction PO, where P is the intersection of an ellipse [eta] with the hyperbola [beta]; and with this velocity the ellipse [eta] can be swimming in the liquid, without distortion for an instant.
At infinity
m m U = - --- e^(-a) cos [beta] = - ----- cos [beta], c a - b
m m V = - --- e^(-a) sin [beta] = - ----- sin [beta], (9) c a + b
a and b denoting the semi-axes of the ellipse [alpha]; so that the liquid is streaming at infinity with velocity Q = m/(a + b) in the direction of the asymptote of the hyperbola [beta].
An ellipse interior to [eta] = [alpha] will move in a direction opposite to the exterior current; and when [eta] = 0, U = [oo], but V = (m/c) sh [alpha] sin [beta].
Negative values of [eta] must be interpreted by a streaming motion on a parallel plane at a level slightly different, as on a double Riemann sheet, the stream passing from one sheet to the other across a cut SS´ joining the foci S, S´. A diagram has been drawn by Col. R. L. Hippisley.
The components of the liquid velocity q, in the direction of the normal of the ellipse [eta] and hyperbola [xi], are
-mJ^(-1)sh([eta] - [alpha]) cos([xi] - [beta]), mJ^(-1)ch([eta] - [alpha]) sin ([xi] - [beta]). (10)
The velocity q is zero in a corner where the hyperbola [beta] cuts the ellipse [alpha]; and round the ellipse [alpha] the velocity q reaches a maximum when the tangent has turned through a right angle, and then
[root](ch 2[alpha] - cos 2[beta]) q = Qe^a ---------------------------------; (11) sh 2[alpha]
and the condition can be inferred when cavitation begins.
With [beta] = 0, the stream is parallel to x0, and
[phi] = m ch([eta] - [alpha])cos [xi]
= -Uc ch([eta] - [alpha])sh [eta] cos [xi]/sh([eta] - [alpha]) (12)
over the cylinder [eta], and as in (12) § 29,
[phi]1 = -Ux = -Uc ch [eta] cos [xi], (13)
for liquid filling the cylinder; and
[phi] th [eta] ------ = --------------------, (14) [phi]1 th ([eta] - [alpha])
over the surface of [eta]; so that parallel to Ox, the effective inertia of the cylinder [eta], displacing M´ liquid, is increased by M´th [eta]/th([eta]- [alpha]), reducing when [alpha] = [oo] to M´th [eta] = M´(b/a).
Similarly, parallel to Oy, the increase of effective inertia is M´/th [eta] th([eta] - [alpha]), reducing to M´/th [eta] = M´(a/b), when [alpha] = [oo], and the liquid extends to infinity.
32. Next consider the motion given by
[phi] = m ch 2([eta] - [alpha]) sin 2[xi], [psi] = -m sh 2([eta] - [alpha]) cos 2[xi]; (1)
in which [psi] = 0 over the ellipse [alpha], and
[psi]´ = [psi] + ½R(x² + y²) = [-m sh 2([eta] - [alpha]) + ¼Rc²] cos 2[xi] + ¼Rc² ch 2[eta], (2)
which is constant over the ellipse [eta] if
¼Rc² = m sh 2([eta] - [alpha]); (3)
so that this ellipse can be rotating with this angular velocity R for an instant without distortion, the ellipse [alpha] being fixed.
For the liquid filling the interior of a rotating elliptic cylinder of cross section
x²/a² + y²/b² = 1, (4)
[psi]1´ = m1(x²/a² + y²/b²) (5)
with
[nabla]²[psi]1´ = -2R = -2m1(1/a² + 1/b²),
[psi]1 = m1(x²/a² + y²/b²) - ½R(x² + y²) = -½R(x² - y²)(a² - b²)/(a² + b²), (6)
[phi]1 = Rxy(a² - b²)/(a² + b²),
w1 = [phi]1 + [psi]1i = -½iR(x + yi)²(a² - b²)/(a² + b²).
The velocity of a liquid particle is thus (a² - b²)/(a² + b²) of what it would be if the liquid was frozen and rotating bodily with the ellipse; and so the effective angular inertia of the liquid is (a² - b²)²/(a² + b²)² of the solid; and the effective radius of gyration, solid and liquid, is given by
k² = ¼(a² + b²), and ¼(a² - b²)²/(a² + b²). (7)
For the liquid in the interspace between [alpha] and [eta],
[phi] m ch 2([eta] - [alpha]) sin 2[xi] ------ = ------------------------------------------- [phi]1 ¼Rc² sh 2[eta] sin 2[xi](a² - b²)/(a² + b²)
= 1/th 2([eta] - [alpha])th 2[eta]; (8)
and the effective k² of the liquid is reduced to
¼c²/th 2([eta] - [alpha]) sh 2[eta], (9)
which becomes ¼c²/sh 2[eta] = 1/8 (a² - b²)/ab, when [alpha] = [oo], and the liquid surrounds the ellipse [eta] to infinity.
An angular velocity R, which gives components -Ry, Rx of velocity to a body, can be resolved into two shearing velocities, -R parallel to Ox, and R parallel to Oy; and then [psi] is resolved into [psi]1 + [psi]2, such that [psi]1 + ½Rx² and [psi]2 + ½Ry² is constant over the boundary.
Inside a cylinder
[phi]1 + [psi]1i = -½iR(x + yi)²a²/(a² + b²), (10)
[phi]2 + [psi]2i = ½iR(x + yi)²b²/(a² + b²), (11)
and for the interspace, the ellipse [alpha] being fixed, and [alpha]1 revolving with angular velocity R
[phi]1 + [psi]1i = -1/8 iRc²sh 2([eta] - [alpha] + [xi]i)(ch 2[alpha] + 1)/sh 2([alpha]1 - [alpha]), (12)
[phi]2 + [psi]2i = 1/8 iRc²sh 2([eta] - [alpha] + [xi]i)(ch 2[alpha] - 1)/sh 2([alpha]1 - [alpha]), (13)
satisfying the condition that [psi]1 and [psi]2 are zero over [eta] = [alpha], and over [eta] = [alpha]1
[psi]1 + ½Rx² = 1/8 Rc²(ch 2[alpha]1 + 1), (14)
[psi]2 + ½Ry² = 1/8 Rc²(ch 2[alpha]1 - 1), (15)
constant values.
In a similar way the more general state of motion may be analysed, given by
w = m ch 2([zeta] - [gamma]), [gamma] = [alpha] + [beta]i, (16)
as giving a homogeneous strain velocity to the confocal system; to which may be added a circulation, represented by an additional term m[zeta] in w.
Similarly, with
x + yi = c[root][sin ([xi] + [eta]i)] (17)
the function
[psi] = Qc sh ½([eta] - [alpha]) sin ½([xi] - [beta]) (18)
will give motion streaming past the fixed cylinder [eta] = [alpha], and dividing along [xi] = [beta]; and then
x² - y² = c² sin [xi] ch [eta], 2xy = c² cos [xi] sh [eta]. (19)
In particular, with sh [alpha] = 1, the cross-section of [eta] = [alpha] is
x^4 + 6x²y² + y^4 = 2c^4, or x^4 + y^4 = c^4 (20)
when the axes are turned through 45°.
33. _Example 3._--Analysing in this way the rotation of a rectangle filled with liquid into the two components of shear, the stream function [psi]1 is to be made to satisfy the conditions
(i.) [nabla]²[psi]1 = 0,
(ii.) [psi]1 + ½Rx² = ½Ra², or [psi]1 = 0 when x = ±a,
(iii.) [psi]1 + ½Rx² = ½Ra², [psi]1 = ½R(a² - x²), when y = ± b.
Expanded in a Fourier series,
32 __ cos (2n + 1) ½[pi]x/a a² - x² = ----- a² \ ---------------------, (1) [pi]³ /__ (2n + 1)³
so that
16 __ cos (2n + 1) ½[pi]x/a · ch(2n + 1) ½[pi]y/a) [psi]1 = ----- a² \ ---------------------------------------------, [pi]³ /__ (2n + 1)^3 · ch(2n + 1) ½[pi]b/a
16 __ cos (2n + 1) ½[pi]z/a w1 = [phi]1 + [psi]1i = iR ----- \ ------------------------------, (2) [pi]³ /__ (2n + 1)^3 ch(2n + 1) ½[pi]b/a
an elliptic-function Fourier series; with a similar expression for [psi]2 with x and y, a and b interchanged; and thence [psi] = [psi]1 + [psi]2.
_Example 4._--Parabolic cylinder, axial advance, and liquid streaming past.
The polar equation of the cross-section being
r^½ cos ½[theta] = a^½, or r + x = 2a, (3)
the conditions are satisfied by
[psi]´ = Ur sin [theta] - 2Ua^½ r^½ sin ½[theta] = 2Ur^½ sin ½[theta](r^½ cos ½[theta] - a^½), (4)
[psi] = 2Ua^½ r^½ sin ½[theta] = -U[root][2a(r-x)], (5)
w = -2Ua^½ z^½, (6)
and the resistance of the liquid is 2[pi][rho]aV²/2g.
A relative stream line, along which [psi]´ = Uc, is the quartic curve
(4a²y² - (y - c)^4 4a²y² + (y-c)^4 y - c = [root][2a(r - x)], x = -------------------, r = ---------------, (7) (4a(y - c)² 4a(y - c)²
and in the absolute space curve given by [psi],
dy (y - c)² 2ac -- = - --------, x = - ----- 2a log (y - c). (8) dx 2ay y - c
34. _Motion symmetrical about an Axis._--When the motion of a liquid is the same for any plane passing through Ox, and lies in the plane, a function [psi] can be found analogous to that employed in plane motion, such that the flux across the surface generated by the revolution of any curve AP from A to P is the same, and represented by 2[pi]([psi] - [psi]0); and, as before, if d[psi] is the increase in [psi] due to a displacement of P to P´, then k the component of velocity normal to the surface swept out by PP´ is such that 2[pi]d[psi] = 2[pi]yk.PP´; and taking PP´ parallel to Oy and Ox,
u = -d[psi]/ydy, v = d[psi]/ydx, (1)
and [psi] is called after the inventor, "Stokes's stream or current function," as it is constant along a stream line (_Trans. Camb. Phil. Soc._, 1842; "Stokes's Current Function," R. A. Sampson, _Phil. Trans._, 1892); and d[psi]/yds is the component velocity across ds in a direction turned through a right angle forward.
In this symmetrical motion
d / 1 d[psi] \ d / 1 d[psi]\ [xi] = 0, [eta] = 0, 2[zeta] = -- ( --- ------ ) + -- ( --- ------ ) dx \ y dx / dy \ y dy /
1 /d²[psi] d²[psi] 1 d[psi]\ 1 = --- ( ------- + ------- - --- ------ ) = - ---[nabla]²[psi], (2) y \ dx² dy² y dy / y
suppose; and in steady motion,
dH 1 d[psi] dH 1 d[psi] -- + --- ----- [nabla]²[psi] = 0, -- + --- ------ [nabla]²[psi] = 0, (3) dx y² dx dy y² dy
so that
2[zeta]/y = -y^(-2)[nabla]²[psi] = dH/d[psi] (4)
is a function of [psi], say [f]´([psi]), and constant along a stream line;
dH/dv = 2q[zeta], H - [f]([psi]) = constant, (5)
throughout the liquid.
When the motion is irrotational,
d[phi] 1 d[psi] d[phi] 1 d[psi] [zeta] = 0, u = - ------ = - --- ------, v = - ------ = --- ------, (6) dx y dy dy y dx
d²[psi] d²[psi] 1 d[psi] [nabla]²[psi] = 0, or ------- + ------- - --- ------ = 0. (7) dx² dy² y dy
Changing to polar coordinates, x = r cos[theta], y = r sin[theta], the equation (2) becomes, with cos[theta] = [mu],
d²[psi] d²[psi] r²------- + (1 - [mu]²) ------- = 2[zeta]r³ sin [theta], (8) dr² d[mu]²
of which a solution, when [zeta] = 0, is
/ B \ dPn / B \ dPn [psi] = ( Ar^(n+1) + --- ) (1 - [mu]²) ----- = ( Ar^(n-1) + ------- ) y²-----, (9) \ r^n / d[mu] \ r^(n+2) / d[mu]
[phi] = {(n + 1)Ar^n - nBr^(-n-1)} Pn, (10)
where Pn denotes the zonal harmonic of the nth order; also, in the exceptional case of
[psi] = A0 cos[theta], [phi] = A0/r;
[psi] = B0r, [phi] = -B0 log tan ½[theta] = -½B0 sh(-1) x/y. (11)
Thus cos[theta] is the Stokes' function of a point source at O, and PA - PB of a line source AB.
The stream function [psi] of the liquid motion set up by the passage of a solid of revolution, moving with axial velocity U, is such that
1 d[psi] dy --- ------ = -U --, [psi] + ½Uy² = constant, (12) y ds ds
over the surface of the solid; and [psi] must be replaced by [psi]´ = [psi] + ½Uy² in the general equations of steady motion above to obtain the steady relative motion of the liquid past the solid.
For instance, with n = 1 in equation (9), the relative stream function is obtained for a sphere of radius a, by making it
[psi]´ = [psi] + ½Uy² = ½U(r² - a³/r) sin² [theta], [psi] = -½Ua³ sin² [theta]/r; (13)
and then
[phi]´ = Ux(1 + ½a³/r²), [phi] = ½Ua³ cos [theta]/r², (14)
d[phi] a³ d[phi] a³ - ------ = U -- cos [theta], - --------- = ½U -- sin [theta], (15) dr r³ rd[theta] r³
so that, if the direction of motion makes an angle [psi] with Ox,
tan ([psi] - [theta]) = ½ tan [theta], tan [psi] = 3 tan [theta]/(2 - tan² [theta]), (16)
Along the path of a liquid particle [psi]´ is constant, and putting it equal to ½Uc²,
(r² - a³/r) sin² [theta] = c², sin² [theta] = c²r/(r³ - a³), (17)
the polar equation; or
y² = c²r³/(r³ - a³), r³ = a³y²/(y² - c²), (18)
a curve of the 10th degree (C10).
In the absolute path in space
cos [psi] = (2 - 3 sin² [theta])/[root](4 - sin² [theta]), and sin³ [theta] = (y³ - c²y)/a³, (19)
which leads to no simple relation.
The velocity past the surface of the sphere is
1 d[psi]´ / a³ \ sin² [theta] ------------ ------- = ½U ( 2r + -- ) ------------- = 3/2 U sin [theta], when r = a; (20) r sin[theta] dr \ r² / r sin [theta]
so that the loss of head is
(9/4 sin² [theta] - 1) U²/2g, having a maximum 5/4 U²/2g, (21)
which must be less than the head at infinite distance to avoid cavitation at the surface of the sphere.
With n = 2, a state of motion is given by
[psi] = -½Uy²a^4[mu]/r^4, [psi]´ = ½Uy²(1 - a^4[mu]/r^4), (22)
[phi]´ = Ux + [phi], [phi] = -1/3 U(a^4/r³)P2, P2 = 3/2 [mu]² - ½, (23)
representing a stream past the surface r^4 = a^4[mu].
35. A circular vortex, such as a smoke ring, will set up motion symmetrical about an axis, and provide an illustration; a half vortex ring can be generated in water by drawing a semicircular blade a short distance forward, the tip of a spoon for instance. The vortex advances with a certain velocity; and if an equal circular vortex is generated coaxially with the first, the mutual influence can be observed. The first vortex dilates and moves slower, while the second contracts and shoots through the first; after which the motion is reversed periodically, as if in a game of leap-frog. Projected perpendicularly against a plane boundary, the motion is determined by an equal opposite vortex ring, the optical image; the vortex ring spreads out and moves more slowly as it approaches the wall; at the same time the molecular rotation, inversely as the cross-section of the vortex, is seen to increase. The analytical treatment of such vortex rings is the same as for the electro-magnetic effect of a current circulating in each ring.
36. _Irrotational Motion in General._--Liquid originally at rest in a singly-connected space cannot be set in motion by a field of force due to a single-valued potential function; any motion set up in the liquid must be due to a movement of the boundary, and the motion will be irrotational; for any small spherical element of the liquid may be considered a smooth solid sphere for a moment, and the normal pressure of the surrounding liquid cannot impart to it any rotation.
The kinetic energy of the liquid inside a surface S due to the velocity function [phi] is given by _ _ _ _ _ / / / | /d[phi]\² /d[phi]\² /d[phi]\² | T = ½[rho] | | | | ( ------ ) + ( ------ ) + ( ------ ) | dx dy dz, _/_/_/ |_ \ dx / \ dy / \ dz / _| _ _ / / d[phi] = ½[rho] | | [phi] ------ dS (1) _/_/ d[nu]
by Green's transformation, d[nu] denoting an elementary step along the normal to the exterior of the surface; so that d[phi]/d[nu] = 0 over the surface makes T = 0, and then
/d[phi]\² /d[phi]\² /d[phi]\² d[phi] d[phi] d[phi] ( ------ ) + ( ------ ) + ( ------ ) = 0, ------ = 0, ------ = 0, ------ = 0 (2) \ dx / \ dy / \ dz / dx dy dz
If the actual motion at any instant is supposed to be generated instantaneously from rest by the application of pressure impulse over the surface, or suddenly reduced to rest again, then, since no natural forces can act impulsively throughout the liquid, the pressure impulse [~[omega]] satisfies the equations
1 d[~omega] 1 d[~omega] 1 d[~omega] ----- --------- = -u, ----- --------- = -v, ----- --------- = -[~omega], (3) [rho] dx [rho] dy [rho] dz
[~omega] = [rho][phi] + a constant, (4)
and the constant may be ignored; and Green's transformation of the energy T amounts to the theorem that the work done by an impulse is the product of the impulse and average velocity, or half the velocity from rest.
In a multiply connected space, like a ring, with a multiply valued velocity function [phi], the liquid can circulate in the circuits independently of any motion of the surface; thus, for example,
[phi] = m[theta] = m tan^(-1) y/x (5)
will give motion to the liquid, circulating in any ring-shaped figure of revolution round Oz.
To find the kinetic energy of such motion in a multiply connected space, the channels must be supposed barred, and the space made acyclic by a membrane, moving with the velocity of the liquid; and then if k denotes the cyclic constant of [phi] in any circuit, or the value by which [phi] has increased in completing the circuit, the values of [phi] on the two sides of the membrane are taken as differing by k, so that the integral over the membrane _ _ _ _ / / d[phi] / / d[phi] | | [phi] ------ dS = k | | ------ dS, (6) _/_/ d[nu] _/_/ d[nu]
and this term is to be added to the terms in (1) to obtain the additional part in the kinetic energy; the continuity shows that the integral is independent of the shape of the barrier membrane, and its position. Thus, in (5), the cyclic constant k = 2[pi]m.
In plane motion the kinetic energy per unit length parallel to Oz _ _ _ _ _ _ _ _ / / | /d[phi]\² /d[phi]\² | / / | /d[psi]\² /d[psi]\² | T = ½[rho] | | | ( ------ ) + ( ------ ) | dx dy = ½[rho] | | | ( ------ ) + ( ------ ) | dx dy _/_/ |_ \ dx / \ dy / _| _/_/ |_ \ dx / \ dy / _| _ _ / d[phi] / d[phi] = ½[rho] | [phi] ------ ds = ½[rho] | [psi] ------ ds. (7) _/ d[nu] _/ d[nu]
For example, in the equilateral triangle of (8) § 28, referred to coordinate axes made by the base and height,
[psi]´ = -2R[alpha][beta][gamma]/h = -½Ry[(h - y)² - 3x²]/h (8)
[psi] = [psi]´ - ½R [(1/3h - y)² + x²]
= -½R [½h³ + 1/3 h²y + h) (x² - y²) - 3x²y + y³] /h (9)
and over the base y = 0,
dx/d[nu] = -dx/dy = + ½R(1/3 h² - 3x²)/h, [psi] = -½R(1/9 h² + x²). (10)
Integrating over the base, to obtain one-third of the kinetic energy T, _ / h/[root]3 1/3 T = ½[rho] | ¼R²(3x^4 - 1/27 h^4) dx/h _/ -h/[root]3
= [rho]R²h^4/135[root]3 (11)
so that the effective k² of the liquid filling the triangle is given by
k² = T/½[rho]R²A = 2h²/45
= 2/5 (radius of the inscribed circle)², (12)
or two-fifths of the k² for the solid triangle.
Again, since
d[phi]/d[nu] = d[psi]/ds, d[phi]/ds = -d[psi]/d[nu], (13) _ _ / / T = ½[rho] | [phi] d[psi] = -½[rho] | [psi] d[phi]. (14) _/ _/
With the Stokes' function [psi] for motion symmetrical about an axis. _ _ / d[psi] / T = ½[rho] | [phi] ------ 2[pi]y ds = [pi][rho] | [phi] d[psi]. (15) _/ yds _/
37. _Flow, Circulation, and Vortex Motion._--The line integral of the tangential velocity along a curve from one point to another, defined by _ _ / / dx dy dz \ / | ( u-- + v-- + w-- ) ds = | (u dx + v dy + z dz), (1) _/ \ ds ds ds / _/
is called the "flux" along the curve from the first to the second point; and if the curve closes in on itself the line integral round the curve is called the "circulation" in the curve.
With a velocity function [phi], the flow _ / - | d[phi] = [phi]1 - [phi]2, (2) _/
so that the flow is independent of the curve for all curves mutually reconcilable; and the circulation round a closed curve is zero, if the curve can be reduced to a point without leaving a region for which [phi] is single valued.
If through every point of a small closed curve the vortex lines are drawn, a tube is obtained, and the fluid contained is called a _vortex filament_.
By analogy with the spin of a rigid body, the component spin of the fluid in any plane at a point is defined as the circulation round a small area in the plane enclosing the point, divided by twice the area. For in a rigid body, rotating about Oz with angular velocity [zeta], the circulation round a curve in the plane xy is _ / / dy dx \ | [zeta] ( x -- - y -- ) ds = [zeta] times twice the area. (3) _/ \ ds ds /
In a fluid, the circulation round an elementary area dxdy is equal to
/ dv \ / du \ / dv du \ udx + ( v + --dx )dy - ( u + --dy )dx - vdy = ( -- - -- )dx dy, (4) \ dx / \ dy / \ dx dy /
so that the component spin is
/ dv du \ ½ ( -- - -- ) = [zeta], (5) \ dx dy /
in the previous notation of § 24; so also for the other two components [xi] and [eta].
Since the circulation round any triangular area of given aspect is the sum of the circulation round the projections of the area on the coordinate planes, the composition of the components of spin, [xi], [eta], [zeta], is according to the vector law. Hence in any infinitesimal part of the fluid the circulation is zero round every small plane curve passing through the vortex line; and consequently the circulation round any curve drawn on the surface of a vortex filament is zero.
If at any two points of a vortex line the cross-section ABC, A´B´C´ is drawn of the vortex filament, joined by the vortex line AA´, then, since the flow in AA´ is taken in opposite directions in the complete circuit ABC AA´B´C´ A´A, the resultant flow in AA´ cancels, and the circulation in ABC, A´B´C´ is the same; this is expressed by saying that at all points of a vortex filament [omega][alpha] is constant where [alpha] is the cross-section of the filament and [omega] the resultant spin (W. K. Clifford, _Kinematic_, book iii.).
So far these theorems on vortex motion are kinematical; but introducing the equations of motion of § 22,
Du dQ Dv dQ Dw dQ -- + -- = 0, -- + -- = 0, -- + -- = 0, (6) dt dx dt dy dt dz _ / Q = | dp/[rho] + V, (7) _/
and taking dx, dy, dz in the direction of u, v, w, and
dx : dy : dz = u : v : w,
D / \ Du D dx -- (u dx + v dy + w dz ) = -- dx + u ---- + ... = -dQ + ½dq², (8) dt \ / dt dt
and integrating round a closed curve _ D / -- | (u dx + v dy + w dz) = 0, (9) dt _/
and the circulation in any circuit composed of the same fluid particles is constant; and if the motion is differential irrotational and due to a velocity function, the circulation is zero round all reconcilable paths. Interpreted dynamically the normal pressure of the surrounding fluid on a tube cannot create any circulation in the tube.
The circulation being always zero round a small plane curve passing through the axis of spin in vortical motion, it follows conversely that a vortex filament is composed always of the same fluid particles; and since the circulation round a cross-section of a vortex filament is constant, not changing with the time, it follows from the previous kinematical theorem that [alpha][omega] is constant for all time, and the same for every cross-section of the vortex filament.
A vortex filament must close on itself, or end on a bounding surface, as seen when the tip of a spoon is drawn through the surface of water.
Denoting the cross-section [alpha] of a filament by dS and its mass by dm, the quantity [omega]dS/dm is called the _vorticity_; this is the same at all points of a filament, and it does not change during the motion; and the vorticity is given by [omega] cos[epsilon]dS/dm, if dS is the oblique section of which the normal makes an angle [epsilon] with the filament, while the aggregate vorticity of a mass M inside a surface S is _ / M^(-1) | [omega] cos [epsilon] dS. _/
Employing the equation of continuity when the liquid is homogeneous,
/ d[zeta] d[eta]\ d² d² d² 2( ------ - ------ ) = [nabla]²u, ... , [nabla]² = - --- - --- - ---, (10) \ dy dz / dx² dy² dz²
which is expressed by
[nabla]²(u,v,w) = 2 curl ([xi], [eta], [zeta]), ([xi], [eta], [zeta]) = ½ curl (u, v, w). (11)
38. _Moving Axes in Hydrodynamics._--In many problems, such as the motion of a solid in liquid, it is convenient to take coordinate axes fixed to the solid and moving with it as the movable trihedron frame of reference. The components of velocity of the moving origin are denoted by U, V, W, and the components of angular velocity of the frame of reference by P, Q, R; and then if u, v, w denote the components of fluid velocity in space, and u´, v´, w´ the components relative to the axes at a point (x, y, z) fixed to the frame of reference, we have
u = U + u´ - yR + zQ, (1) v = V + v´- zP + xR, w = W + w´ - xQ + yP.
Now if k denotes the component of absolute velocity in a direction fixed in space whose direction cosines are l, m, n,
k = lu + mv + nw; (2)
and in the infinitesimal element of time dt, the coordinates of the fluid particle at (x, y, z) will have changed by (u´, v´, w´)dt; so that
Dk dl dm dn -- = --u + --v + --w dt dt dt dt
/ du du du du \ + l( -- + u´-- + v´-- + w´-- ) \ dt dx dy dz /
/ dv dv dv dv \ + m( -- + u´-- + v´-- + w´-- ) \ dt dx dy dz /
/ dw dw dw dw \ + n( -- + u´-- + v´-- + w´-- ). (3) \ dt dx dy dz /
But as l, m, n are the direction cosines of a line fixed in space,
dl dm dn -- = mR - nQ, -- = nP - lR, -- = lQ - mP; (4) dt dt dt
so that
Dk / du du du du \ -- = l( -- - vR + wQ + u´-- + v´-- + w´-- ) + m(...) + n(...) dt \ dt dx dy dz /
/ 1 dp \ / 1 dp \ / 1 dp \ = l( X- --- -- ) + m( Y - --- -- ) + n( Z - --- -- ), (5) \ p dx / \ p dy / \ p dz /
for all values of l, m, n, leading to the equations of motion with moving axes.
When the motion is such that
d[phi] d[psi] d[phi] d[psi] d[phi] d[psi] u = - ------ - m------, v = - ------ - m------, w = - ------ - m------, (6) dx dx dy dy dz dz
as in §25 (1), a first integral of the equations in (5) may be written _ / dp d[phi] d[psi] / d[phi] d[psi] \ | ----- + V + ½q² - ------ - m------ + (u - u´) ( ------ + m------ ) _/ [rho] dt dt \ dx dx /
/ d[phi] d[psi] \ / d[phi] d[psi] \ + (v - v´)( ------ + m------ ) + (w - w´)( ------ + m------ ) = F(t), (7) \ dy dy / \ dz dz / in which
d[phi] d[phi] d[phi] d[phi] ------ - (u - u´)------ - (v -v´)------ - (w - w´)------ dt dx dy dz
d[phi] d[phi] d[phi] d[phi] = ------ - (U - yR + zQ)------ - (V - zP + xR)------ - (W - xQ + yP)------ (8) dt dx dy dz
is the time-rate of change of [phi] at a point fixed in space, which is left behind with velocity components u - u´, v - v´, w - w´.
In the case of a steady motion of homogeneous liquid symmetrical about Ox, where O is advancing with velocity U, the equation (5) of § 34
p/[rho] + V + ½q´² - [f]([psi]´) = constant (9)
becomes transformed into
p U d[psi] ----- + V + ½q² - --- ------ + ½U² - [f]([psi] + ½Uy²) = constant, (10) [rho] y dy
[psi]´ = [psi] + ¼U², (11)
subject to the condition, from (4) §34,
y^(-2)[nabla]²[psi]´ = -[f]´([psi]´), y^(-2)[nabla]²[psi] = -[f]´([psi] + ½Uy²). (12)
Thus, for example, with
[psi]´ = ¾Uy²(r²a^(-2) - 1), r² = x² + y², (13)
for the space inside the sphere r = a, compared with the value of [psi]´ in §34 (13) for the space outside, there is no discontinuity of the velocity in crossing the surface.
Inside the sphere
d / 1 d[psi]´\ d / 1 d[psi]´\ 15 y 2[zeta] = --- ( --- ------- ) + --- ( --- ------- ) = ---U ---, (14) dx \ y dx / dy \ y dy / 2 a²
so that §34 (4) is satisfied, with
15 15 [f]´([psi]´)= ---Ua^(-2), [f]([psi]´) = ---U[psi]´a^(-2); (15) 2 2
and (10) reduces to _ _ p 9 | / x² \² / y² \² | ----- + V - ---U | ( --- -1 ) - ( --- - ½ ) | = constant; (16) [rho] 8 |_ \ a² / \ a² / _|
this gives the state of motion in M. J. M. Hill's spherical vortex, advancing through the surrounding liquid with uniform velocity.
39. As an application of moving axes, consider the motion of liquid filling the ellipsoidal case
x² y² z² --- + --- + --- = 1; (1) a² b² c²
and first suppose the liquid to be frozen, and the ellipsoid to be rotating about the centre with components of angular velocity [xi], [eta], [zeta]; then
u = - y[zeta] + z[eta], v = - z[xi] + x[zeta], w = - x[eta] + y[xi]. (2)
Now suppose the liquid to be melted, and additional components of angular velocity [Omega]1, [Omega]2, [Omega]3 communicated to the ellipsoidal case; the additional velocity communicated to the liquid will be due to a velocity-function
b² - c² c² - a² a² - b² [phi] = - [Omega]1 ------- yz - [Omega]2 -------zx - [Omega]3 -------xy, (3) b² + c² c² + a² a² + b²
as may be verified by considering one term at a time.
If u´, v´, w´ denote the components of the velocity of the liquid relative to the axes,
2a² 2a² u´ = u + yR - zQ = ------- [Omega]3 y - ------- [Omega]2 z, (4) a² + b² c² + a²
2b² 2b² v´ = v + zP - xR = ------- [Omega]1 z - ------- [Omega]3 x, (5) b² + c² a² + b²
2c² 2c² w´ = w + xQ - yP = ------- [Omega]2 x - ------- [Omega]1 y, (6) c² + a² b² + c²
P = [Omega]1 + [xi], Q = [Omega]2 + [eta], R = [Omega]3 + [zeta]. (7)
Thus
x y z u´ --- + v´ --- + w´ --- = 0, (8) a2 b2 c2
so that a liquid particle remains always on a similar ellipsoid.
The hydrodynamical equations with moving axes, taking into account the mutual gravitation of the liquid, become
1 dp du du du du ----- -- + 4[pi][rho]Ax + -- - vR + wQ + u´-- + v´-- + w´-- = 0, ... , ... , (9) [rho] dx dt dx dy dz
where _ / [oo] abcd[lambda] A, B, C = | ---------------------------------------------- _/ 0 (a² + [lambda], b² + [lambda], c² + [lambda])P
P² = 4(a² + [lambda]) (b² + [lambda]) (c² + [lambda]). (10)
With the values above of u, v, w, u´, v´, w´, the equations become of the form
1 dp ----- -- + 4[pi][rho]Ax + [alpha]x + hy + gz = 0, (11) [rho] dx
1 dp ----- -- + 4[pi][rho]By + hx + [beta]y + fz = 0, (12) [rho] dy
1 dp ----- -- + 4[pi][rho]Cz + gx + fy + [gamma]z = 0, (13) [rho] dz
and integrating
p[rho]^(-1) + 2[pi][rho](Ax² + By² + Cz²)
+ ½([alpha]x² + [beta]y² + [gamma]z² + 2fyz + 2gzx + 2hxy) = const., (14)
so that the surfaces of equal pressure are similar quadric surfaces, which, symmetry and dynamical considerations show, must be coaxial surfaces; and f, g, h vanish, as follows also by algebraical reduction; and
4c²(c² - a²) / c² - a² \² [alpha] = ------------[Omega]2² - ( -------[Omega]2 - [eta] ) (c² + a²)² \ c² + a² /
4b²(a² - b²) / a² - b² \² - ------------[Omega]3² - ( -------[Omega]3 - [zeta] ), (15) (a² + b²)² \ a² + b² /
with similar equations for [beta] and [gamma].
If we can make
(4[pi][rho]A + [alpha])x² = (4[pi][rho]B + [beta])b² = (4[pi][rho]C + [gamma])c², (16)
the surfaces of equal pressure are similar to the external case, which can then be removed without affecting the motion, provided [alpha], [beta], [gamma] remain constant.
This is so when the axis of revolution is a principal axis, say Oz; when
[Omega]1 = 0, [Omega]2 = 0, [xi] = 0, [eta] = 0. (17)
If [Omega]3 = 0 or [theta]3 = [zeta] in addition, we obtain the solution of Jacobi's ellipsoid of liquid of three unequal axes, rotating bodily about the least axis; and putting a = b, Maclaurin's solution is obtained of the rotating spheroid.
In the general motion again of the liquid filling a case, when a = b, [Omega]3 may be replaced by zero, and the equations, hydrodynamical and dynamical, reduce to
d[xi] 2c² d[eta] 2a² ----- = - -------[Omega]2 [zeta], ------ = -------[Omega]1 [zeta], dt a² + c² dt a² + c²
d[zeta] 2c² ------- = -------([Omega]2 [xi] - [Omega]2 [eta]) (18) dt a² + c²
d[Omega]1 a² + c² --------- = [Omega]2 [zeta] + -------[eta][zeta], dt a² - c²
d[Omega]2 a² + c² --------- = [Omega]1 [zeta] + -------[xi][zeta]; (19) dt a² - c²
of which three integrals are
a² [xi]² + [eta]² = L - --[zeta]², (20) c²
(a² + c²)² [Omega]1² + [Omega]2² = M + ------------ [zeta]², (21) 2c²(a² - c²)
a² + c² [Omega]1 [xi] + [Omega]2 [eta]N = + ------- [zeta]²; (22) 4c² and then
/ d[zeta]\² 4c^4 ( ------- ) = --------- ([Omega]2[xi] - [Omega]1²[eta])² \ dt / (a² + c²)
4c^4 = ---------- [([xi]² + [eta]²)([Omega)1² + [Omega]2²) -([Omega]1[xi] + [Omega]2[eta])²] (a² + c²)² _ 4c^4 | / (a² + c²)² a² (a² + c²)\ = ---------- | LM - N² + ( L------------ - M --- - N --------- ) [zeta]² (a² + c²)² |_ \ 2c²(a² + c²) c² 2c² / _ (a² + c²)(9a² - c²) | - ------------------- [zeta]^4 | = Z, (23) 16c^4(a² - c²) _|
where Z is a quadratic in [zeta]², so that [zeta] is an elliptic function of t, except when c = a, or 3a.
Put [Omega]1 = [Omega] cos [phi], [Omega]2 = -[Omega] sin [phi],
d[phi] d[Omega]1 d[Omega]2 (a² + c²) [Omega]2 ------ = -----------[Omega]2 - [Omega]1 --------- = [Omega]²[zeta] - ---------([Omega]1 [xi] + [Omega]2 [eta])[zeta], (24) dt dt dt (a² - c²)
a² + c² N + ------- d[phi] (a² + c²) 4c² ------ = [zeta] - --------- · -------------------------, (25) dt (a² - c²) (a² + c²)² M + ------------[zeta]² 2c²(a² - c²)
a² + c² _ _ N + -------[zeta]² / [zeta]d a² + c² / 4c² [zeta] d[zeta] [phi] = | ------- - ------- | ------------------------ · --------------, (26) _/ [root]Z a² - c² _/ (a² + c²)² [root]Z M + ------------[zeta]² 2c²(a² - c²)
which, as Z is a quadratic function of [zeta]², are non-elliptic integrals; so also for [psi], where [xi] = [omega] cos [psi], [eta] = -[omega] sin [psi].
In a state of steady motion
d[zeta] [Omega]1 [Omega]2 ------- = 0, -------- = --------, (27) dt [xi] [eta]
[phi] = [psi] = nt, suppose, (28)
[Omega]1[xi] + [Omega]2 [eta] = [Omega][omega], (29)
d[phi] a² + c² [omega] ------ = [zeta]- ------- -------[zeta], (30) dt a² - c² [Omega]
d[psi] 2a² [Omega] ------ = - ------- -------[zeta], (31) dt a² + c² [omega]
a² + c² [omega] 2a² [Omega] 1 - ------- ------- = - ------- -------, (32) a² - c² [Omega] a² + c² [omega]
/ [omega] a² + c² \² (a² - c²)(9a² - c²) ( ------- - ½ ------- ) = -------------------, (33) \ [Omega] a² - c² / 4(a² + c²)
and a state of steady motion is impossible when 3a > c > a.
An experiment was devised by Lord Kelvin for demonstrating this, in which the difference of steadiness was shown of a copper shell filled with liquid and spun gyroscopically, according as the shell was slightly oblate or prolate. According to the theory above the stability is regained when the length is more than three diameters, so that a modern projectile with a cavity more than three diameters long should fly steadily when filled with water; while the old-fashioned type, not so elongated, would be highly unsteady; and for the same reason the gas bags of a dirigible balloon should be over rather than under three diameters long.
40. _A Liquid Jet._--By the use of the complex variable and its conjugate functions, an attempt can be made to give a mathematical interpretation of problems such as the efflux of water in a jet or of smoke from a chimney, the discharge through a weir, the flow of water through the piers of a bridge, or past the side of a ship, the wind blowing on a sail or aeroplane, or against a wall, or impinging jets of gas or water; cases where a surface of discontinuity is observable, more or less distinct, which separates the running stream from the dead water or air.
Uniplanar motion alone is so far amenable to analysis; the velocity function [phi] and stream function [psi] are given as conjugate functions of the coordinates x, y by
w = [f](z) where z = x + yi, w = [phi] + [psi]i, (1)
and then
dw d[phi] d[psi] -- = ------ + i------ = -u + vi; (2) dz dx dx
so that, with u = q cos [theta], v = q sin [theta], the function
dz Q Q Q [zeta] = -Q -- = -------- = ---(u + vi) = --- (cos [theta] + i sin [theta]), (3) dw (u - vi) q² q
gives [zeta] as a vector representing the reciprocal of the velocity q in direction and magnitude, in terms of some standard velocity Q.
To determine the motion of a jet which issues from a vessel with plane walls, the vector [zeta] must be Constructed so as to have a constant direction [theta] along a plane boundary, and to give a constant skin velocity over the surface of a jet, where the pressure is constant.
It is convenient to introduce the function
[Omega] = log [zeta] = log(Q/q) + [theta]i (4)
so that the polygon representing [Omega] conformally has a boundary given by straight lines parallel to the coordinate axes; and then to determine [Omega] and w as functions of a variable u (not to be confused with the velocity component of q), such that in the conformal representation the boundary of the [Omega] and w polygon is made to coincide with the real axis of u.
It will be sufficient to give a few illustrations.
Consider the motion where the liquid is coming from an infinite distance between two parallel walls at a distance xx´ (fig. 4), and issues in a jet between two edges A and A´; the wall xA being bent at a corner B, with the external angle [beta] = ½[pi]/n.
The theory of conformal representation shows that the motion is given by _ _ | [root](b - a´·u - a) + [root](b - a·u - a´) |^1/n [zeta] = | ------------------------------------------- | , u = ae^(-[pi]w/m); (5) |_ [root](a - a´·u - b) _|
where u = a, a´ at the edge A, A¹; u = b at a corner B; u = 0 across xx´ where [phi] = [oo]; and u = [oo], [phi] = [oo] across the end JJ´ of the jet, bounded by the curved lines APJ, A´P´J´, over which the skin velocity is Q. The stream lines xBAJ, xA´J´ are given by [psi] = 0, m; so that if c denotes the ultimate breadth JJ´ of the jet, where the velocity may be supposed uniform and equal to the skin velocity Q,
m = Qc, c = m/Q.
If there are more B corners than one, either on xA or x´A´, the expression for [zeta] is the product of corresponding factors, such as in (5).
Restricting the attention to a single corner B,
/ Q \^n [root](b - a´·u - a) + [root](b - a·u - a´) [zeta]^n = ( --- ) (cos n[theta] + i sin n[theta] = -------------------------------------------, (6) \ q / [root](a - a´.u - b)
/ Q \^n / Q \^n ch n[omega] = ch log( --- ) cos n[theta] + i sh log ( --- ) sin n[theta] \ q / \ q /
/b - a´ /u - a´ = ½([zeta]^n + [zeta]^(-n)) = / ------ / ------ (7) \/ a - a´ \/ u - b´
/ Q \ / Q \^n sh n[Omega] = sh log ( --- ) cos n[theta] + i ch log ( --- ) sin n[theta] \ q / \ q /
/b - a´ /u - a´ = ½([zeta]^n - [zeta]^(-n)) = / ------ / ------ (8) \/ a - a´ \/ u - b´
[oo] > a > b > 0 > a´ > -[oo] (9)
and then
d[Omega] 1 [root](b - a´·b - a´) dw m -------- = - -- ---------------------------, -- = - ------ (10) du 2n (u - b)[root](a - a·u - a´) du [pi]u´
the formulas by which the conformal representation is obtained.
For the [Omega] polygon has a right angle at u = a, a´, and a zero angle at u = b, where [theta] changes from 0 to ½[pi]/n and [Omega] increases by ½i[pi]/n; so that
d[Omega] A [root](b - a·b - a´) -------- = ---------------------------, where A = --------------------. (11) du (u - b)[root](u - a·u - a´) 2n
And the w polygon has a zero angle at u = 0, [oo], where [psi] changes from 0 to m and back again, so that w changes by im, and
dw B m -- = ---, where B = - ----. (12) du u [pi]
Along the stream line xBAPJ,
[psi] = 0, u = ae^(-[pi][phi]/m); (13)
and over the jet surface JPA, where the skin velocity is Q,
d[phi] ------ = -q = -Q, u = ae^([pi]sQ/m) = ae^([pi]s/c), (14) ds
denoting the arc AP by s, starting at u = a;
/b - a´ /u - a´ ch n[Omega] = cos n[theta] = / ----- / ------ (15) \/ a - a´ \/ u - b´
/a - b /u - a´ sh n[Omega] = i sin n[theta] = i / ------ / ------ (16) \/ a - a´ \/ u - b´
[oo] > u = ae^([pi]s/c) > a, (17)
and this gives the intrinsic equation of the jet, and then the radius of curvature
ds 1 d[phi] i dw i dw / d[Omega] [rho] = - -------- = --- -------- = --- ------- = --- -- / ------- d[theta] Q d[theta] Q d[Omega] Q du / du
c u - b [root](u - a·u - a´) = ----·2n----- --------------------, (18) [pi] u [root](a - b·b - a´)
not requiring the integration of (11) and (12)
If [theta] = [alpha] across the end JJ´ of the jet, where u = [oo], q = Q,
/b - a´ /a - b ch n[Omega] = cos n[alpha] = /-------, sh n[Omega] = i sin n[alpha]= / ------, (19) \/ a - a´ \/ a - a´
Then
a - b·b - a´ a - a´ cos 2n[alpha] - cos 2n[theta] = 2------------ = ½ sin² 2n[alpha]------ a - a´·u - b u - b
[root](a - b.b - a´)[root](u - a·u - b´) sin 2n[theta] = 2---------------------------------------- (20) a - a´·u - b
[root](a - a·b - a´) = sin 2n[alpha]--------------------; u - b
2n c / b \ [root](a - b·b - a´) ----- ----- = ( 1 + ----- ) -------------------- (21) [phi] [rho] \ u - b/ [root](u - a·u - a´)
a - a´ + (a + a´) cos 2n[alpha] - [a + a´ + (a - a´) cos 2n[alpha] cos 2n[theta] = -------------------------------------------------------------------------------- (a - a´) sin² 2n[alpha]
cos 2n[alpha] - cos 2n[theta] × ----------------------------- sin 2n[theta]
Along the wall AB, cos n[theta] = 0, sin n[theta] = 1,
a > u > b, (22)
/ Q \^n /b - a´ /a - u ch n[Omega] = i sh log ( --- ) = i / ------ / ------ (23) \ q / \/ a - a´ \/ u - b´
/ Q \^n /a - b /u - a´ sh n[Omega] = i ch log ( --- ) = i / ------ / ------ (24) \ q / \/ a - a´ \/ u - b´
ds ds d[phi] m c Q -- = ------ ------ = ------ = --- -- (25) du d[phi] dt [pi]qu [pi] qu _ AB / a Q du [pi]-- = | --- -- c _/ b q u _ _ _ / | [root](a - b)[root](u - a´) + [root](b - a´)[root](a - u) |^1/n du = | | --------------------------------------------------------- | --. (26) _/ |_ [root](a - a´)[root](u - b´) _| u
Along the wall Bx, cos n[theta] = 1, sin n[theta] = 0,
b > u > 0 (27)
/ Q \^n /b - a´ /a - u ch n[Omega] = ch log ( --- ) = / ------ / ------ (28) \ q / \/ a - a´ \/ b - u´
/ Q \^n /a - b /u - a´ sh n[Omega] = sh log ( --- ) = / ------ / ------. (29) \ q / \/ a - a´ \/ b - u
At x where [phi] = [oo], u = 0, and q = q0,
/ Q \^n /b - a´ / a /a - b / -a´ ( --- ) = / ------ / --- + / ------ / ---. (30) \ q0 / \/ a - a´ \/ b \/ a - a´ \/ q
In crossing to the line of flow x´A´P´J´, [psi] changes from 0 to m, so that with q = Q across JJ´, while across xx´ the velocity is q0, so that
m = q0.xx´ = Q.JJ´ (31) _ _ JJ´ q0 | /b - a´ / a /a - b / -a´ |^1/n -- = --- = | / ------ / --- - / ------ / --- | , (32) xx´ Q |_ \/ a - a´ \/ b \/ a - a´ \/ q _|
giving the contraction of the jet compared with the initial breadth of the stream.
Along the line of flow x´A´P´J´, [psi] = m, u = a´e^(-[pi][phi]/m), and from x´ to A´, cos n[theta] = 1, sin n[theta] = 0,
/ Q \^n /b - a´ /a - u ch n[Omega] = ch log ( --- ) = / ------ / ------, (33) \ q / \/ a - a´ \/ b - u´
/ Q \^n /a - b /u - a´ sh n[Omega] = sh log ( --- ) = / ------ / ------. (34) \ q / \/ a - a´ \/ b - u´
0 > u > a´. (35)
Along the jet surface A´J´, q = Q,
/b - a´ /a - u ch n[Omega] = cos n[theta] = / ------ / ------, (36) \/ a - a´ \/ b - u´
/a - b /a´ - u sh n[Omega] = i sin n[theta] = i / ------ / ------, (37) \/ a - a´ \/ b - u´
a´ > u = a´e^([pi]/sc) > -[oo], (38)
giving the intrinsic equation.
41. The first problem of this kind, worked out by H. v. Helmholtz, of the efflux of a jet between two edges A and A1 in an infinite wall, is obtained by the symmetrical duplication of the above, with n = 1, b = 0, a´ = -[oo], as in fig. 5,
/u - a / -a ch [Omega] = / -----, sh [Omega] = / ---; (1) \/ u \/ u
and along the jet APJ, [oo] > u = ae^([pi]s/c) > a,
sh [Omega] = i sin [theta] - i[root](a/u) = ie^(-½[pi]s/c), (2) _ _ / [oo] / c c PM = | sin [theta] ds = | e^(-½[pi]s/c) ds = ----- e^(-½[pi]s/c) = ----- sin [theta], (3) _/ s _/ ½[pi] ½[pi]
so that PT = c/½[pi], and the curve AP is the tractrix; and the coefficient of contraction, or
breadth of the jet [pi] ---------------------- = --------. (4) breadth of the orifice [pi] + 2
A change of [Omega] and [theta] into n[Omega] and n[theta] will give the solution for two walls converging symmetrically to the orifice AA1 at an angle [pi]/n. With n = ½, the reentrant walls are given of Borda's mouthpiece, and the coefficient of contraction becomes ½. Generally, by making a´ = - [oo], the line x´A´ may be taken as a straight stream line of infinite length, forming an axis of symmetry; and then by duplication the result can be obtained, with assigned n, a, and b, of the efflux from a symmetrical converging mouthpiece, or of the flow of water through the arches of a bridge, with wedge-shaped piers to divide the stream.
42. Other arrangements of the constants n, a, b, a´ will give the results of special problems considered by J. M. Michell, _Phil. Trans._ 1890.
Thus with a´ = 0, a stream is split symmetrically by a wedge of angle [pi]/n as in Bobyleff's problem; and, by making a = [oo], the wedge extends to infinity; then
/ b / n ch n[Omega] = / -----, sh n[Omega] = / -----. (1) \/ b - u \/ b - u
Over the jet surface [psi] = m, q = Q,
u = - e^([pi][phi]/m) = - be^([pi]²/c),
/ 1 / e^([pi]²/c) ch [Omega] = cos n[theta] = / ---------------, sh [Omega] = i sin n[theta] = i / ---------------, (2) \/ e^([pi]²/c) + 1 \/ e^([pi]²/c) + 1
½[pi] ds 2n e^(½[pi]²/c) = tan n[theta], ----- -------- = -------------. (3) c d[theta] sin 2n[theta]
For a jet impinging normally on an infinite plane, as in fig. 6, n = 1,
e^(½[pi]²/c) = tan [theta], ch (½[pi]s/c) sin 2[theta] = 1, (4)
sh ½[pi]x/c = cot [theta], sh ½[pi]y/c = tan [theta],
sh ½[pi]x/c sh ½[pi]y/c = 1, e^(½[pi](x + y)/c) = e^(½[pi]x/c) + e^(½[pi]y/c) + 1. (5)
With n = ½, the jet is reversed in direction, and the profile is the catenary of equal strength.
In Bobyleff's problem of the wedge of finite breadth,
/ b /u - a /b - a / u ch n[Omega] = / --- / -----, sh n[Omega] = / ----- / -----, (6) \/ a \/ u - b \/ a \/ u - b
/ b /a - b cos n[alpha] = / ---, sin n[alpha] = / -----, (7) \/ a \/ a
and along the free surface APJ, q = Q, [psi] = 0, u = e^(-[pi][phi]/m) = ae^([pi]s/c),
/ e^([pi]²/c) - 1 cos n[theta] = cos n[alpha] / ---------------------------, \/ e^([pi]²/c) - cos² n[alpha]
cos² n[alpha] sin² n[theta] e^([pi]²/c) = -----------------------------, (8) sin² n[theta] - sin² n[alpha]
the intrinsic equation, the other free surface A´P´J´ being given by
cos² n[alpha] sin² n[theta] e^([pi]²/c) = -----------------------------. (9) sin² n[alpha] - sin² n[theta]
Putting n = 1 gives the case of a stream of finite breadth disturbed by a transverse plane, a particular case of Fig. 7.
When a = b, [alpha] = 0, and the stream is very broad compared with the wedge or lamina; so, putting w = w´(a - b)/a in the penultimate case, and
u = ae^(-w) [asympt] a - (a - b)w´, (10)
/w´ + 1 / 1 ch n[Omega] = / ------, sh n[Omega] = / --------, (11) \/ w´ \/ [root]w´
in which we may write
w´ = [phi] + [psi]i. (12)
Along the stream line xABPJ, [psi] = 0; and along the jet surface APJ, -1 > [phi] > -[oo]; and putting [phi] = -[pi]s/c - 1, the intrinsic equation is
[pi]s/c = cot² n[theta], (13)
which for n = 1 is the evolute of a catenary.
43. When the barrier AA´ is held oblique to the current, the stream line xB is curved to the branch point B on AA´ (fig. 7), and so must be excluded from the boundary of u; the conformal representation is made now with
d[Omega] [root](b - a·b - a´) -------- = - ---------------------------- (1) du (u - b) [root](u - a·u - a´)
dw m 1 m´ 1 m + m´ u - b -- = - ---- ----- - ---- -----, = - ------ · -------------, du [pi] u - j [pi] u - j [pi] u - j·u - j´
mj´ + m´j b = ---------, (2) m + m´
taking u = [oo] at the source where [phi] = [oo], u = b at the branch point B, u = j, j´ at the end of the two diverging streams where [phi] = -[oo]; while [psi] = 0 along the stream line which divides at B and passes through A, A´; and [psi] = m, -m´ along the outside boundaries, so that m/Q, m´/Q is the final breadth of the jets, and (m + m´)/Q is the initial breadth, c1 of the impinging stream. Then
/b - a´ /u - a /b - a /u - a´ ch ½[Omega] = / ------ / -----, sh ½[Omega] = / ------ / ------, (3) \/ a - a´ \/ u - b \/ a - a´ \/ u - b
2b - a - a´ N ch [Omega] = ----------- - -----, a - a´ u - b
/ [root](2·a - u·u - a´) sh [Omega] = / N----------------------, \/ u - b
a - b·b - a´ N = 2------------. (4) a - a´
Along a jet surface, q = Q, and
ch[Omega] = cos [theta] = cos [alpha] - ½sin² [alpha](a - a´)/(u - b), (5)
if [theta] = [alpha] at the source x of the jet xB, where u = [oo]; and supposing [theta] = [beta], [beta]´ at the end of the streams where u = j, j´,
u - b ½ sin² [alpha] u - j cos[theta] - cos[beta] ----- = -------------------------, ------ = ½ sin² [alpha]-----------------------------------------------------, a - a´ cos [alpha] - cos [theta] a - a´ (cos [alpha] - cos [beta])(cos [alpha] - cos [theta])
u - j´ cos [theta] - cos [beta]´ ----- = ½ sin² [alpha]------------------------------------------------------; (6) a - a´ (cos [alpha] - cos [beta]´)(cos [alpha] - cos [theta])
and [psi] being constant along a stream line
d[phi] dw ds d[phi] dw du ------ = --, Q -------- = -------- = -- --------, du du d[theta] d[theta] du d[theta]
[pi]Q ds [pi] ds (cos [alpha] - cos [beta])(cos [alpha] - cos [beta]´)sin[theta] ------ -------- = ---- -------- = ---------------------------------------------------------------------------------, m + m´ d[theta] c d[theta] (cos [alpha] - cos [theta])(cos [theta] - cos [beta])(cos [theta] - cos [alpha]´)
sin [theta] cos [alpha] - cos [beta]´ sin [theta] = ------------------------- + ------------------------- · ------------------------ cos [alpha] - cos [theta] cos [beta] - cos [beta]´ cos [theta] - cos [beta]
cos [alpha] - cos [beta] sin [theta] ------------------------ · -------------------------, (7) cos [beta] - cos [beta]´ cos [theta] - cos [beta]´
giving the intrinsic, equation of the surface of a jet, with proper attention to the sign.
From A to B, a > u > b, [theta] = 0,
Q a - a´ ch [Omega] = ch log --- = cos [alpha] - ½ sin² [alpha] ------ q a - b
Q [root](a - u·u - a´) sh [Omega] = sh log --- = -------------------- sin [alpha] q u - b
Q (u - b) cos [alpha] - ½(a - a´) sin² [alpha] + [root](a - u·u - a´)sin[alpha] --- = ----------------------------------------------------------------------------- (8) q u - b
ds ds d[phi] Q dw Q -- = Q ------ ------ = - --- -- du d[phi] du q du
m + m´ (u - b) cos [alpha] - ½(a - a´) sin² [alpha] + [root](a - u·u - a´) sin [alpha] = ------ · ------------------------------------------------------------------------------- (9) [pi] j - u·u - j´ _ AB / a (2b - a - a´)(u - b) - 2(a - b)(b - a´) + 2[root](a - b·b -a´·a - u·u -a´) [pi]-- = | -------------------------------------------------------------------------- du, (10) c _/ b a - a´·j - u·u - j´
with a similar expression for BA´.
The motion of a jet impinging on an infinite barrier is obtained by putting j = a, j´ = a´; duplicated on the other side of the barrier, the motion reversed will represent the direct collision of two jets of unequal breadth and equal velocity. When the barrier is small compared with the jet, [alpha] = [beta] = [beta]´, and G. Kirchhoff's solution is obtained of a barrier placed obliquely in an infinite stream.
Two corners B1 and B2 in the wall xA, with a´ = -[oo], and n = 1, will give the solution, by duplication, of a jet issuing by a reentrant mouthpiece placed symmetrically in the end wall of the channel; or else of the channel blocked partially by a diaphragm across the middle, with edges turned back symmetrically, problems discussed by J. H. Michell, A. E. H. Love and M. Réthy.
When the polygon is closed by the walls joining, instead of reaching back to infinity at xx´, the liquid motion must be due to a source, and this modification has been worked out by B. Hopkinson in the _Proc. Lond. Math. Soc._, 1898.
Michell has discussed also the hollow vortex stationary inside a polygon (_Phil. Trans._, 1890); the solution is given by
ch n[Omega] = sn w, sh n[Omega] = i cn w (11)
so that, round the boundary of the polygon, [psi] = K´, sin n[theta] = 0; and on the surface of the vortex [psi] = 0, q = Q, and
cos n[theta] = sn [phi], n[theta] = ½[pi] - am s/c, (12)
the intrinsic equation of the curve.
This is a closed Sumner line for n = 1, when the boundary consists of two parallel walls; and n = ½ gives an Elastica.
44. _The Motion of a Solid through a Liquid._--An important problem in the motion of a liquid is the determination of the state of velocity set up by the passage of a solid through it; and thence of the pressure and reaction of the liquid on the surface of the solid, by which its motion is influenced when it is free.
Beginning with a single body in liquid extending to infinity, and denoting by U, V, W, P, Q, R the components of linear and angular velocity with respect to axes fixed in the body, the velocity function takes the form
[phi] = U_[phi]1 + V_[phi]2 + W_[phi]3 + P_[chi]1 + Q_[chi]2 + R_[chi]3, (1)
where the [phi]'s and [chi]'s are functions of x, y, z, depending on the shape of the body; interpreted dynamically, C - [rho][phi] represents the impulsive pressure required to stop the motion, or C + [rho][phi] to start it again from rest.
The terms of [phi] may be determined one at a time, and this problem is purely kinematical; thus to determine [phi]1, the component U alone is taken to exist, and then l, m, n, denoting the direction cosines of the normal of the surface drawn into the exterior liquid, the function [phi]1 must be determined to satisfy the conditions
(i.) [nabla]²[phi]1 = 0. throughout the liquid;
(ii.) d[ph]1/d[upsilon] = -l, the gradient of [phi] down the normal at the surface of the moving solid;
(iii.) d[ph]1/d[upsilon] = 0, over a fixed boundary, or at infinity;
similarly for [phi]2 and [phi]3.
To determine [chi]1 the angular velocity P alone is introduced, and the conditions to be satisfied are
(i.) [nabla]²[chi]1 = 0, throughout the liquid;
(ii.) d[chi]1/d[upsilon] = mz - ny, at the surface of the moving body, but zero over a fixed surface, and at infinity; the same for [chi]2 and [chi]3.
For a cavity filled with liquid in the interior of the body, since the liquid inside moves bodily for a motion of translation only,
[phi]1 = -x, [phi]2 = -y, [phi]3 = -z; (2)
but a rotation will stir up the liquid in the cavity, so that the [chi]'s depend on the shape of the surface.
The ellipsoid was the shape first worked out, by George Green, in his _Research on the Vibration of a Pendulum in a Fluid Medium_ (1833); the extension to any other surface will form an important step in this subject.
A system of confocal ellipsoids is taken
x² y² z² ------------- + ------------- + ------------- = 1 (3) a² + [lambda] b² + [lambda] c² + [lambda]
and a velocity function of the form
[phi] = x[psi], (4)
where [psi] is a function of [lambda] only, so that [psi] is constant over an ellipsoid; and we seek to determine the motion set up, and the form of [psi] which will satisfy the equation of continuity.
Over the ellipsoid, p denoting the length of the perpendicular from the centre on a tangent plane,
px py pz l = -------------, m = -------------, n = ------------- (5) a² + [lambda] b² + [lambda] c² + [lambda]
p²x² p²y² p²z² 1 = ---------------- + ---------------- + ----------------, (6) (a² + [lambda])² (b² + [lambda])² (c² + [lambda])²
p² = (a² + [lambda])l² + (b² + [lambda])m² + (c² + [lambda])n², (7) = a²l² + b²m² + c²n² + [lambda],
dp d[lambda] 2p-- = ---------; (8) ds ds
Thence
d[phi] dx d[psi] ------ = --[psi] + x------ ds ds ds
dx d[psi] dp = --[psi] + 2(a² + [lambda])--------- l--, (9) ds d[lambda] ds
so that the velocity of the liquid may be resolved into a component -[psi] parallel to Ox, and -2(a² + [lambda])l d[psi]/d[lambda] along the normal of the ellipsoid; and the liquid flows over an ellipsoid along a line of slope with respect to Ox, treated as the vertical.
Along the normal itself
d[phi] / d[psi] \ ----- = ( [psi] + 2(a² + [lambda])-------- )l, (10) ds \ d[lambda] /
so that over the surface of an ellipsoid where [lambda] and [psi] are constant, the normal velocity is the same as that of the ellipsoid itself, moving as a solid with velocity parallel to Ox
d[psi] U = -[psi] - 2(a² + [lambda])---------, (11) d[lambda]
and so the boundary condition is satisfied; moreover, any ellipsoidal surface [lambda] may be supposed moving as if rigid with the velocity in (11), without disturbing the liquid motion for the moment.
The continuity is secured if the liquid between two ellipsoids [lambda] and [lambda]1, moving with the velocity U and U1 of equation (11), is squeezed out or sucked in across the plane x = 0 at a rate equal to the integral flow of the velocity [psi] across the annular area [alpha]1 - [alpha] of the two ellipsoids made by x = 0; or if _ / [lambda]1 d[alpha] [alpha]U - [alpha]1U1 = | [psi]-------- d[lambda], (12) _/ [lambda] d[lambda]
[alpha] = [pi][root](b² + [lambda]·c² + [lambda]). (13)
Expressed as a differential relation, with the value of U from (11), _ _ d | d[psi] | d[alpha] --------- | [alpha][psi] + 2(a² + [lambda])[alpha]--------- | - [psi]--------- = 0, (14) d[lambda] |_ d[lambda] _| d[lambda]
d[psi] d / d[psi] \ 3[alpha]--------- + 2(a² + [lambda])-------- ( [alpha]-------- ) = 0, (15) d[lambda] d[lambda] \ d[lambda]/
and integrating
d[psi] (a² + [lambda])^3/2 [alpha]-------- = a constant, (16) d[lambda]
so that we may put _ / Md[lambda] [psi] = | ----------------, (17) _/ (a² + [lambda])P
P² = 4(a² + [lambda])(b² + [lambda])(c² + [lambda]), (18)
where M denotes a constant; so that [psi] is an elliptic integral of the second kind.
The quiescent ellipsoidal surface, over which the motion is entirely tangential, is the one for which
d[psi] 2(a² + [lambda])--------- + [psi] = 0, (19) d[lambda]
and this is the infinite boundary ellipsoid if we make the upper limit [lambda]1 = [oo].
The velocity of the ellipsoid defined by [lambda] = 0 is then
d[psi]0 U = -2a²--------- - [psi]0 d[lambda] _ M / [oo] Md[lambda] = --- - | ---------------- abc _/ 0 (a² + [lambda])P
M = --- (1 - A0), (20) abc
with the notation _ / [oo] abc d[lambda] A or A_[lambda] = | ---------------- _/[lambda] (a² + [lambda])P _ d / [oo] d[lambda] = -2abc--- | ---------, (21) da²_/ [lambda] P
so that in (4)
M UxA xA_[lambda] [phi] = ---xA = ------, [phi]1 = -----------, (22) abc 1 - A0 1 - A0
in (1) for an ellipsoid.
The impulse required to set up the motion in liquid of density [rho] is the resultant of an impulsive pressure [rho][phi] over the surface S of the ellipsoid, and is therefore _ _ _ _ / / / / | | [rho][phi]l dS = [rho][psi]0 | | xl dS _/_/ _/_/
= [rho][psi]0 (volume of the ellipsoid) = [psi]0 W´, (23)
where W´ denotes the weight of liquid displaced.
Denoting the effective inertia of the liquid parallel to Ox by [alpha]W´. the momentum
[alpha]W´U = [psi]0W´ (24)
[psi]0 A0 [alpha] = ------ = ------; (25) U 1 - A0
in this way the air drag was calculated by Green for an ellipsoidal pendulum.
Similarly, the inertia parallel to Oy and Oz is
B0 C0 [beta]W´ = ------ W´, [gamma]W´ = ------ W´, (26) 1 - B0 1 - C0 _ / [oo] abc d[lambda] B_[lambda], C_[lambda] = | -------------------------------; (27) _/[lambda] (b² + [lambda], c² + [lambda])P
and
A + B + C = abc/½P, A0 + B0 + C0 = 1. (28)
For a sphere
a = b = c, A0 = B0 = C0 = 1/3, [alpha] = [beta] = [gamma] = ½, (29)
so that the effective inertia of a sphere is increased by half the weight of liquid displaced; and in frictionless air or liquid the sphere, of weight W, will describe a parabola with vertical acceleration
W - W´ ------- g. (30) W + ½W´
Thus a spherical air bubble, in which W/W´ is insensible, will begin to rise in water with acceleration 2g.
45. When the liquid is bounded externally by the fixed ellipsoid [lambda] = [lambda]1, a slight extension will give the velocity function [phi] of the liquid in the interspace as the ellipsoid [lambda] = 0 is passing with velocity U through the confocal position; [phi] must now take the form x([psi] + N), and will satisfy the conditions in the shape _ abc /[lambda]1 abcd[lambda] ------ + | ---------------- A + B1 + C1 a1b1c1 _/ [lambda] (a² + [lambda])P [phi] = Ux ----------------- = Ux ------------------------------------------, (1) B0 + C0 - B1 - C1 abc /[lambda]1 abcd[lambda] 1 - ------ - | ---------------- a1b1c1 _/0 (a² + [lambda])P
and any confocal ellipsoid defined by [lambda], internal or external to [lambda] = [lambda]1, may be supposed to swim with the liquid for an instant, without distortion or rotation, with velocity along Ox
B_[lambda] + C_[lambda] - B1 - C1 U ---------------------------------- B0 + C0 - B1 - C1
Since - Ux is the velocity function for the liquid W´ filling the ellipsoid [lambda] = 0, and moving bodily with it, the effective inertia of the liquid in the interspace is
A0 + B1 + C1 ----------------- W´. (2) B0 + C0 - B1 - C1
If the ellipsoid is of revolution, with b = c,
A + 2B1 [phi] = ½Ux -------, (3) B0 - B1
and the Stokes' current function [psi] can be written down
B - B1 [psi] = - ½Uy² -------; (4) B0 - B1
reducing, when the liquid extends to infinity and B1 = 0, to A B [phi] = ½Ux --, [psi] = - ½Uy² --; (5) B0 B0
so that in the relative motion past the body, as when fixed in the current U parallel to xO,
/ A \ / B \ [phi]´ = ½Ux ( 1 + -- ), [psi]´ = ½Uy² ( 1 - -- ). (6) \ B0 / \ B0 /
Changing the origin from the centre to the focus of a prolate spheroid, then putting b² = pa, [lambda] = [lambda]´a, and proceeding to the limit where a = [oo], we find for a paraboloid of revolution
p B p B = ½ -------------, -- = -------------, (7) p + [lambda]´ B0 p + [lambda]´
y² ------------- = p + [lambda]´ - 2x, (8) p + [lambda]´
with [lambda]´ = 0 over the surface of the paraboloid; and then
[psi]´ = ½U [y² - p[root](x² + y²) + px]; (9)
[psi] = -½Up [[root](x² + y²) - x]; (10)
[phi] = -½Up log [ [root](x² + y²) + x]. (11)
The relative path of a liquid particle is along a stream line
[psi]´ = ½Uc², a constant, (12)
p²y² - (y² - c²)² p²y² - (y² - c²)² x = -----------------, [root](x² + y²) = ----------------- (13) 2p(y² - c²) 2p(y² - c²)
a C4; while the absolute path of a particle in space will be given by
dy r - x y² - c² -- = - ----- = -------, (14) dx y 2py
y² - c² = a²e^(-x/p). (15)
46. Between two concentric spheres, with
a² + [lambda] = r², a² + [lambda]1 = a1², (1)
A = B = C = a³/3r³,
a³ a³ a³ a³ -- + 2 --- -- + 2 --- r³ a1³ r³ a1³ [phi] = ½Ux -----------, [psi] = ½Uy² ----------; (2) 1 - a^4/a1² 1 - a³/a1³
and the effective inertia of the liquid in the interspace is
A0 + 2A1 a1³ + 2a³ --------- W´ = ½ --------- W´. (3) 2A0 - 2A1 a1³ - a³
When the spheres are not concentric, an expression for the effective inertia can be found by the method of images (W. M. Hicks, _Phil. Trans._, 1880).
The image of a source of strength [mu] at S outside a sphere of radius a is a source of strength [mu]a/[f] at H, where OS = [f], OH = a²/f, and a line sink reaching from the image H to the centre O of line strength - [mu]/a; this combination will be found to produce no flow across the surface of the sphere.
Taking Ox along OS, the Stokes' function at P for the source S is [mu] cos PSx, and of the source H and line sink OH is [mu](a/[f]) cos PHx and -([mu]/a)(PO - PH); so that
/ a PO - PH \ [psi] = [mu] (cos PSx + --- cos PHx - ------- ), (4) \ [f] a /
and [psi] = -[mu], a constant, over the surface of the sphere, so that there is no flow across.
When the source S is inside the sphere and H outside, the line sink must extend from H to infinity in the image system; to realize physically the condition of zero flow across the sphere, an equal sink must be introduced at some other internal point S´.
When S and S´ lie on the same radius, taken along Ox, the Stokes' function can be written down; and when S and S´ coalesce a doublet is produced, with a doublet image at H.
For a doublet at S, of moment m, the Stokes' function is
d y² m-- cos PSx = -m---; (5) df PS³
and for its image at H the Stokes' function is
d a³ y² m-- cos PHx = -m-- ---; (6) df f³ PH³
so that for the combination
/a³ 1 1 \ y² / a³ f³\ [psi] = my² ( -- --- - --- ) = m-- ( --- - --- ), (7) \f³ PH³ PS³/ f³ \PH³ PS³/
and this vanishes over the surface of the sphere.
There is ao Stokes' function when the axis of the doublet at S does not pass through O; the image system will consist of an inclined doublet at H, making an equal angle with OS as the doublet S, and of a parallel negative line doublet, extending from H to O, of moment varying as the distance from O.
A distribution of sources and doublets over a moving surface will enable an expression to be obtained for the velocity function of a body moving in the presence of a fixed sphere, or inside it.
The method of electrical images will enable the stream function [psi]´ to be inferred from a distribution of doublets, finite in number when the surface is composed of two spheres intersecting at an angle [pi]/m, where m is an integer (R. A. Herman, _Quart. Jour. of Math._ xxii.).
Thus for m = 2, the spheres are orthogonal, and it can be verified that
/ a1³ a2³ a³ \ [psi]´ = ½Uy² ( 1 - --- - --- + -- ), (8) \ r1³ r2³ r³ /
where a1, a2, a = a1a2/[root](a1² + a2²) is the radius of the spheres and their circle of intersection, and r1, r2, r the distances of a point from their centres.
The corresponding expression for two orthogonal cylinders will be
/ a1² a2² a² \ [psi]´ = Uy ( 1 - --- - --- + -- ). (9) \ r1² r2² r² /
With a2 = [oo], these reduce to
/ a^5 \ x / a^4 \ x [psi]´ = ½Uy² ( 1 - --- ) ---, or Uy ( 1 - --- ) ---, (10) \ r^5 / a \ r^4 / a
for a sphere or cylinder, and a diametral plane.
Two equal spheres, intersecting at 120°, will require _ _ | x a³ a^4(a - 2x) a³ a^4(a + 2x) | [psi]´ = ½Uy² | --- - ---- + ----------- + ---- - ----------- |, (11) |_ a 2r1³ 2r1^5 2r2³ 2r2^5 _|
with a similar expression for cylinders; so that the plane x = 0 may be introduced as a boundary, cutting the surface at 60°. The motion of these cylinders across the line of centres is the equivalent of a line doublet along each axis.
47. The extension of Green's solution to a rotation of the ellipsoid was made by A. Clebsch, by taking a velocity function
[phi] = xy[chi] (1)
for a rotation R about Oz; and a similar procedure shows that an ellipsoidal surface [lambda] may be in rotation about Oz without disturbing the motion if
/ 1 1 \ dx ( ------------ + ------------ ) [chi] + 2--------- \ a² + [lambda] b² + [lambda] / d[lambda] R = - -----------------------------------------------------, (2) 1/(b² + [lambda] - 1/(a² = [lambda])
and that the continuity of the liquid is secured if
d[chi] (a² + [lambda])^3/2 (b² + [lambda])^3/2 (c² + [lambda]) ½--------- = constant, (3) d[lambda] _ / [oo] Nd[lambda] N B_[lambda] - A_[lambda] [chi] = | ------------------------------- = --- . -----------------------; (4) _/[lambda] (a² + [lambda])(b² + [lambda])P abc a² - b²
and at the surface [lambda] = 0,
/ 1 1\ N B0 - A0 N 1 ( -- + -- ) --- ------- - --- ---- \a² b²/ abc a² - b² abc a²b² R = - ----------------------------------, (5) 1/b² - 1/a²
N 1/b² - 1/a² --- = R --------------------------, (6) abc 1 / 1 1\ B0 - A0 ---- - ( -- + -- ) ------- a²b² \a² b²/ a² - b²
(a² - b²)²/(a² + b²) = R -------------------------------. (a² - b²)/(a² + b²) - (B0 - A0)
The velocity function of the liquid inside the ellipsoid [lambda] = 0 due to the same angular velocity will be
[phi]1 = Rxy(a² - b²)/(a² + b²), (7)
and on the surface outside
N B0 - A0 [phi]0 = xy[chi]0 = xy--- -------, (8) abc a² - b²
so that the ratio of the exterior and interior value of [phi] at the surface is
[phi]0 B0 - A0 ------ = -------------------------------, (9) [phi]1 (a² - b²)/(a² + b²) - (B0 - A0)
and this is the ratio of the effective angular inertia of the liquid, outside and inside the ellipsoid [lambda] = 0.
The extension to the case where the liquid is bounded externally by a fixed ellipsoid [lambda] = [lambda]1 is made in a similar manner, by putting
[phi] = xy([chi] + M), (10)
and the ratio of the effective angular inertia in (9) is changed to
a1² - b1² abc (B0 - A0) - (B1 - A1) + --------- ------ a1² + b1² a1b1c1 --------------------------------------------------. (11) a² - b² a1² - b1² abc ------- - --------- ------ - (B0 - A0) + (B1 - A1) a² + b² a1² + b1² a1b1c1
Make c = [oo] for confocal elliptic cylinders; and then _ /[oo] ab ab / /b² + [lambda] \ A[lambda] = | ----------------------------------------------------- = ------- ( 1 - / ------------- ), (12) _/[lambda] (a² + [lambda])[root]([4·a² + [lambda]·b² + [lambda]) a² - b² \ \/ a² + [lambda] /
ab / /a² + [lambda] \ B[lambda] = ------- ( / ------------- - 1 ), C[lambda]= 0; a² - b² \ \/ b² + [lambda] /
and then as above in § 31, with
a = c ch [alpha], b = c sh [alpha], a1 = [root](a² + [lambda]) = c ch [alpha]1, b1 = c sh [alpha]1 (13)
the ratio in (11) agrees with § 31 (6).
As before in § 31, the rotation may be resolved into a shear-pair, in planes perpendicular to Ox and Oy.
A torsion of the ellipsoidal surface will give rise to a velocity function of the form [phi] = xyz[Omega], where [Omega] can be expressed by the elliptic integrals A_[lambda], B_[lambda], C_[lambda], in a similar manner, since _ / [oo] [Omega] = L | d[lambda]/P³ _/ [lambda]
48. The determination of the [phi]'s and [chi]'s is a kinematical problem, solved as yet only for a few cases, such as those discussed above.
But supposing them determined for the motion of a body through a liquid, the kinetic energy T of the system, liquid and body, is expressible as a quadratic function of the components U, V, W, P, Q, R. The partial differential coefficient of T with respect to a component of velocity, linear or angular, will be the component of momentum, linear or angular, which corresponds.
Conversely, if the kinetic energy T is expressed as a quadratic function of x1, x2, x3, y1, y2, y3, the components of momentum, the partial differential coefficient with respect to a momentum component will give the component of velocity to correspond.
These theorems, which hold for the motion of a single rigid body, are true generally for a flexible system, such as considered here for a liquid, with one or more rigid bodies swimming in it; and they express the statement that the work done by an impulse is the product of the impulse and the arithmetic mean of the initial and final velocity; so that the kinetic energy is the work done by the impulse in starting the motion from rest.
Thus if T is expressed as a quadratic function of U, V, W, P, Q, R, the components of momentum corresponding are
dT dT dT x1 = --, x2 = --, x3 = --, (1) dU dV dW
dT dT dT y1 = --, y2 = --, y3 = --; dP dQ dR
but when it is expressed as a quadratic function of x1, x2, x3, y1, y2, y3,
dT dT dT U = ---, V = ---, W = ---, (2) dx1 dx2 dx3
dT dT dT P = ---, Q = ---, R = ---. dy1 dy2 dy3
The second system of expression was chosen by Clebsch and adopted by Halphen in his _Fonctions elliptiques_; and thence the dynamical equations follow
dx1 dT dT X = --- - x2--- + x3---, Y = ..., Z = ..., (3) dt dy3 dy2
dy1 dT dT dT dT L = --- - y2--- + y3--- - x2--- + x2---, M = ..., N = ..., (4) dt dy3 dy2 dx3 dx2
where X, Y, Z, L, M, N denote components of external applied force on the body.
These equations are proved by taking a line fixed in space, whose direction cosines are l, m, n, then
dl dm dn -- = mR - nQ, -- = nP - lR, -- = lQ - mP. (5) dt dt dt
If P denotes the resultant linear impulse or momentum in this direction
P = lx1 + mx2 + nx3, (6)
dP dl dm dn -- = --x1 + --x2 + --x3 dt dt dt dt
dx1 dx2 dx3 + l--- + m--- + n---, dt dt dt
/ dx1 \ = l ( --- - x2R + x3Q ) \ dt /
/ dx2 \ + m ( --- - x3P + x1R ) \ dt /
/ dx3 \ + n ( --- - x1Q + x2P ) \ dt /
= lX + mY + nZ, (7)
for all values of l, m, n.
Next, taking a fixed origin [Omega] and axes parallel to Ox, Oy, Oz through O, and denoting by x, y, z the coordinates of O, and by G the component angular momentum about [Omega] in the direction (l, m, n)
G = l(y1 - x2z + x3y) + m(y2 - x3x + x1z) + n(y3 - x1y + x2x). (8)
Differentiating with respect to t, and afterwards moving the fixed origin up to the moving origin O, so that
dx dy dz x = y = z = 0, but -- = U, -- = V, -- = W, dt dt dt
dG / dy1 \ -- = l ( --- - y2R + y3Q - x2W + x3V ) dt \ dt /
/ dy2 \ + m ( --- - y3P + y1R - x3U + x1W ) \ dt /
/ dy3 \ + n ( --- - y1Q + y2P - x1V + x2U ) \ dt /
= lL + mM + nN, (9)
for all values of l, m, n.
When no external force acts, the case which we shall consider, there are three integrals of the equations of motion
(i.) T = constant,
(ii.) x1² + x2² + x3² = F², a constant,
(iii.) x1y1 + x2y2 + x3y3 = n = GF, a constant;
and the dynamical equations in (3) express the fact that x1, x2, x3 are the components of a constant vector having a fixed direction; while (4) shows that the vector resultant of y1, y2, y3 moves as if subject to a couple of components
x2W - x3V, x3U - x1W, x1V - x2U, (10)
and the resultant couple is therefore perpendicular to F, the resultant of x1, x2, x3, so that the component along OF is constant, as expressed by (iii).
If a fourth integral is obtainable, the solution is reducible to a quadrature, but this is not possible except in a limited series of cases, investigated by H. Weber, F. Kötter, R. Liouville, Caspary, Jukovsky, Liapounoff, Kolosoff and others, chiefly Russian mathematicians; and the general solution requires the double-theta hyperelliptic function.
49. In the motion which can be solved by the elliptic function, the most general expression of the kinetic energy was shown by A. Clebsch to take the form
T = ½p(x1² + x2²) + ½p´x3² + q(x1y1 + x2y2) + q´x3y3 + ½r(y1² + y2²) + ½r´y3² (1)
so that a fourth integral is given by
dy3/dt = 0, y3 = constant; (2)
dx3 --- = x1(qx2 + ry2) - x2(qx1 + ry1) = r(x1y2 - x2y1), (3) dt
1 / dx3 \² -- ( --- ) = (x1² + x2²)(y1² + y2²) - (x1y1 + x2y2)² r² \ dt /
= (x1² + x2²)(y1² + y2²) - (FG - x3y3)² = (x1² + x2²)(y1² + y2² + y3² - G²) - (Gx3 - Fy3)², (4)
in which
x1² + x2² = F² - x3², x1y1 + x2y2 = FG - x3y3, (5)
r(y1² + y2²) = 2T - p(x1² + x2²) - p´x3² - 2q(x1y1 + x2y2) - 2q´x3y3 - r´y3² = (p - p´)x3² + 2(q - q´)x3y3 + m1, (6)
m1 - 2T - pF² - 2qFG - r1y3² (7)
so that
1 / dx3 \² -- ( --- ) = X3 (8) r² \ dt /
where X3 is a quartic function of x3, and thus t is given by an elliptic integral of the first kind; and by inversion x3 is in elliptic function of the time t. Now
(x1 - x2i)(y1 + y2i) = x1y1 + x2y2 + i(x1y2 - x2y1) = FG - xy3y3 + i[V-]X3, (9)
y1 + y2i FG - x3y3 + i[root]X3 -------- = --------------------- , (10) x1 + x2i x1² + x2²
d -- (x1 + x2i) = -i[(q´ - q)x3 + r´y3] + irx3(y1 + y2i), (11) dt
d FG - x3y3 + i[root]X3 --- log (x1 + x2i) = dti -(q´ - q)x - r´y + rx ---------------------, (12) dti F² - x3²
d /x1 + x2i Fy3 - Gx3 --- log \ / -------- = -(q´ - q)x3 - (r´ - r)y3 - Fr---------, (13) dti \/ x1 - x2i F² - x3²
requiring the elliptic integral of the third kind; thence the expression of x1 + x2i and y1 + y2i.
Introducing Euler's angles [theta], [phi], [psi],
x1 = F sin [theta] sin [phi], x2 = F sin [theta] cos [phi],
x1 + x2i = iF sin [theta][epsilon]^(-[psi]i), x3 = F cos [theta]; (14)
d[psi] sin [theta] ------ = P sin [phi] + Q cos[phi], (15) dt
d[psi] dT dT F sin²[theta] ------ = --- x1 + --- x2 dt dy1 dy2
= (qx1 + ry1)x1 + (qx2 + ry2)x2
= qx1² + x2²) + r (x1y1 + x2y2)
= gF² sin² [theta] + r(FG - x3y3), (16) _ / FG - x3y3 Fr dx3 [psi] - qFt = | --------- --------, (17) _/ F² - x3² [root]X3
elliptic integrals of the third kind.
Employing G. Kirchhoff's expressions for X, Y, Z, the coordinates of the centre of the body, __ __ __ FX = y1 cos xY + y2 cos yY + y3 cos zY, (18) __ __ __ FY = -y1 cos xX + y2 cos yX + y3 cos zX, (19) __ __ __ G = y1 cos xZ + y2 cos yZ + y3 cos zZ, (20)
F²(X² + Y²) = y1² + y2² + y3² - G², (21)
Fy3 - Gx3 + i[root]X3 F(X + Yi) = --------------------- [epsilon]^[psi]_i. (22) [root](F² - x3²)
Suppose x3 - F is a repeated factor of X3, then y3 = G, and _ _ | p´ - p q´ - q | X3 = (x3 - F)² | ------(x3 + F)² + 2------G(x3 + F) - G² |, (23) |_ r r _|
and putting x3 - F = y, _ / dy \² | p´ - p q´ - q ( -- ) = r²y² | 4 ------ F² + 4 ------ FG - G² \ dt / |_ r r _ / p´ - p q´ - q \ p´ - p | + 2 ( 2 ------ F + ------ G ) y + ------ y² |, (24) \ r r / r _|
so that the stability of this axial movement is secured if
p´ - p q´ - q A = 4 ------F² + 4 ------FG - G² (25) r r
is negative, and then the axis makes r[V-](-A)/[pi] nutations per second. Otherwise, if A is positive _ / dy rt = | ---------------------- _/ y[root](A + 2By + Cy²)
1 sh^(-1) [root]A[root](A + 2By + Cy²) 1 ch^(-1) A + By = ------- ---------------------------- = ------- ---------------, (26) [root]A ch^(-1) y[root](B² ~ AC) [root]A sh^(-1) y[root](B² ~ AC)
and the axis falls away ultimately from its original direction.
A number of cases are worked out in the American Journal of Mathematics (1907), in which the motion is made algebraical by the use of the pseudo-elliptic integral. To give a simple instance, changing to the stereographic projection by putting tan ½[theta] = x,
(Nxe[psi]i)^3/2 = (x + 1)[root]X1 + i(x - 1)[root]X2, (27)
X1 -- = ±ax^4 + 2ax³ ± 3(a + b)x² + 2bx ± b, (28) X2
N³ = -8(a + b), (29)
will give a possible state of motion of the axis of the body; and the motion of the centre may then be inferred from (22).
50. The theory preceding is of practical application in the investigation of the stability of the axial motion of a submarine boat, of the elongated gas bag of an airship, or of a spinning rifled projectile. In the steady motion under no force of such a body in a medium, the centre of gravity describes a helix, while the axis describes a cone round the direction of motion of the centre of gravity, and the couple causing precession is due to the displacement of the medium.
In the absence of a medium the inertia of the body to translation is the same in all directions, and is measured by the weight W, and under no force the C.G. proceeds in a straight line, and the axis of rotation through the C.G. preserves its original direction, if a principal axis of the body; otherwise the axis describes a cone, right circular if the body has uniaxial symmetry, and a Poinsot cone in the general case.
But the presence of the medium makes the effective inertia depend on the direction of motion with respect to the external shape of the body, and on W´ the weight of fluid medium displaced.
Consider, for example, a submarine boat under water; the inertia is different for axial and broadside motion, and may be represented by
c1 = W + W´[alpha], c2 = W + W´[beta], (1)
where [alpha], [beta] are numerical factors depending on the external shape; and if the C.G. is moving with velocity V at an angle [phi] with the axis, so that the axial and broadside component of velocity is u = V cos [phi], v = V sin [phi], the total momentum F of the medium, represented by the vector OF at an angle [theta] with the axis, will have components, expressed in sec. lb.,
u V v V F cos [theta] = c1 --- = (W + W´[alpha]) --- cos [phi], F sin [theta] = c2 --- = (W + W´[beta]) --- sin [phi]. (2) g g g g
Suppose the body is kept from turning as it advances; after t seconds the C.G. will have moved from O to O´, where OO´ = Vt; and at O´ the momentum is the same in magnitude as before, but its vector is displaced from OF to O´F´.
For the body alone the resultant of the components of momentum
V V V W --- cos [phi] and W --- sin [phi] is W --- sec. lb., (3) g g g
acting along OO´, and so is unaltered.
But the change of the resultant momentum F of the medium as well as of the body from the vector OF to O´F´ requires an impulse couple, tending to increase the angle FOO´, of magnitude, in sec. foot-pounds
F·OO´·sin FOO´ = FVt sin ([theta] - [phi]), (4)
equivalent to an incessant couple
N = FV sin ([theta] - [phi]) = (F sin [theta] cos [phi] - F cos [theta] sin [phi])V = (c2 - c1)(V²/g) sin [phi] cos [phi] = W´([beta] - [alpha]uv/g). (5)
This N is the couple in foot-pounds changing the momentum of the medium, the momentum of the body alone remaining the same; the medium reacts on the body with the same couple N in the opposite direction, tending when c2-c1 is positive to set the body broadside to the advance.
An oblate flattened body, like a disk or plate, has c2 - c1 negative, so that the medium steers the body axially; this may be verified by a plate dropped in water, and a leaf or disk or rocket-stick or piece of paper falling in air. A card will show the influence of the couple N if projected with a spin in its plane, when it will be found to change its aspect in the air.
An elongated body like a ship has c2-c1 positive, and the couple N tends to disturb the axial movement and makes it unstable, so that a steamer requires to be steered by constant attention at the helm.
Consider a submarine boat or airship moving freely with the direction of the resultant momentum horizontal, and the axis at a slight inclination [theta]. With no reserve of buoyancy W = W´, and the couple N, tending to increase [theta], has the effect of diminishing the metacentric height by h ft. vertical, where
c1 u² Wh tan[theta] = N = (c2 - c1) -- --- tan [theta], (6) c2 g
c2 - c1 c1 u² 1 + [alpha] u² h = ------- -- --- = ([beta] - [alpha] ----------- ---. (7) W c2 g 1 + [beta] g
51. An elongated shot is made to preserve its axial flight through the air by giving it the spin sufficient for stability, without which it would turn broadside to its advance; a top in the same way is made to stand upright on the point in the position of equilibrium, unstable statically but dynamically stable if the spin is sufficient; and the investigation proceeds in the same way for the two problems (see GYROSCOPE).
The effective angular inertia of the body in the medium is now required; denote it by C1 about the axis of the figure, and by C2 about a diameter of the mean section. A rotation about the axis of a figure of revolution does not set the medium in motion, so that C1 is the moment of inertia of the body about the axis, denoted by Wk1². But if Wk2² is the moment of inertia of the body about a mean diameter, and [omega] the angular velocity about it generated by an impulse couple M, and M´ is the couple required to set the surrounding medium in motion, supposed of effective radius of gyration k´,
Wk2²[omega] = M - M´, W´k´²[omega] = M´, (1)
Wk2² + W´k´²[omega] = M, (2)
C2 = Wk2² + W´k´² = (W + W´[epsilon])k2², (3)
in which we have put k´² = [epsilon]k², where [epsilon] is a numerical factor depending on the shape.
If the shot is spinning about its axis with angular velocity p, and is preceding steadily at a rate [mu] about a line parallel to the resultant momentum F at an angle [theta], the velocity of the vector of angular momentum, as in the case of a top, is
C1p[mu] sin [theta] - C2[mu]² sin [theta] cos [theta]; (4)
and equating this to the impressed couple (multiplied by g), that is, to
c1 gN = (c1 - c2)-- u² tan [theta], (5) c2
and dividing out sin[theta], which equated to zero would imply perfect centring, we obtain
c1 C2[mu]² cos [theta] - C1p[mu] + (c2 - c1)-- u² sec [theta] = 0. (6) c2
The least admissible value of p is that which makes the roots equal of this quadratic in [mu], and then
C1 [mu] = ½ --p sec [theta], (7) C2
the roots would be imaginary for a value of p smaller than given by
c1 C1²p² - 4(c2 - c1)-- C2u² = 0, (8) c2
p² c1 C2 -- = 4 (c2 - c1) -- ---. (9) u² c2 C1²
_Table of Rifling for Stability of an Elongated Projectile, x Calibres long, giving [delta] the Angle of Rifling, and n the Pitch of Rifling in Calibres._
+-----------------------+-----------------+-----------------+-----------------+-----------------+ | | Cast-iron Common| Palliser Shell | Solid Steel | Solid Lead | | | Shell [f] = 2/3,| [f] = ½, S.G. 8.| Bullet | Bullet [f] = 0, | | | S.G. 7.2. | | [f] = 0, S.G. 8.| S.G. 10.9. | +--------+--------------+--------+--------+--------+--------+--------+--------+--------+--------+ | x |[beta]-[alpha]|[delta] | n |[delta] | n |[delta] | n |[delta] | n | +--------+--------------+--------+--------+--------+--------+--------+--------+--------+--------+ | 1.0 | 0.0000 | 0° 0´|Infinity| 0° 0´|Infinity| 0° 0´|Infinity| 0° 0´|Infinity| | 2.0 | 0.4942 | 2 49 | 63.87 | 2 32 | 71.08 | 2 29 | 72.21 | 2 08 | 84.29 | | 2.5 | 0.6056 | 3 46 | 47.91 | 3 23 | 53.32 | 3 19 | 54.17 | 2 51 | 63.24 | | 3.0 | 0.6819 | 4 41 | 38.45 | 4 13 | 42.79 | 4 09 | 43.47 | 3 38 | 50.74 | | 3.5 | 0.7370 | 5 35 | 32.13 | 5 02 | 35.75 | 4 58 | 36.33 | 4 15 | 42.40 | | 4.0 | 0.7782 | 6 30 | 27.60 | 5 51 | 30.72 | 5 45 | 31.21 | 4 56 | 36.43 | | 4.5 | 0.8100 | 7 24 | 24.20 | 6 40 | 26.93 | 6 32 | 27.36 | 5 37 | 31.94 | | 5.0 | 0.8351 | 8 16 | 21.56 | 7 28 | 23.98 | 7 21 | 24.36 | 6 18 | 28.44 | | 6.0 | 0.8721 | 10 05 | 17.67 | 9 04 | 19.67 | 8 56 | 19.98 | 7 40 | 23.33 | | 10.0 | 0.9395 | 16 57 | 10.31 | 15 19 | 11.47 | 15 05 | 11.65 | 13 00 | 13.60 | |Infinity| 1.0000 | 90 00 | 0.00 | 90 00 | 0.00 | 90 00 | 0.00 | 90 00 | 0.00 | +--------+--------------+--------+--------+--------+--------+--------+--------+--------+--------+
If the shot is moving as if fired from a gun of calibre d inches, in which the rifling makes one turn in a pitch of n calibres or nd inches, so that the angle [delta] of the rifling is given by
tan [delta] = [pi]d/nd = ½ dp/u, (10)
which is the ratio of the linear velocity of rotation ½dp to u, the velocity of advance,
c1 C2d² tan² [delta] = [pi]²/n² = d²p²/4u² = (c2 - c1) -- ---- c2 C1²
W´ / W´ \ / k2 \² 1 + --- [alpha] ( 1 + ---[epsilon] ) ( -- ) W´ W \ W / \ d / = --- ([beta] - [alpha]) --------------- · -----------------------------. (11) W W´ / k1 \^4 1 + ---[beta] ( -- ) W \ d /
For a shot in air the ratio W´/W is so small that the square may be neglected, and formula (11) can be replaced for practical purpose in artillery by
[pi]² W´ / k2 \² / / k1 \^4 tan²[delta] = ----- = --- ([beta] - [alpha]) ( -- ) / ( -- ), (12) n² W \ d / / \ d /
if then we can calculate [beta], [alpha], or [beta] - [alpha] for the external shape of the shot, this equation will give the value of [delta] and n required for stability of flight in the air.
The ellipsoid is the only shape for which [alpha] and [beta] have so far been determined analytically, as shown already in § 44, so we must restrict our calculation to an egg-shaped bullet, bounded by a prolate ellipsoid of revolution, in which, with b = c, _ _ / [oo] ab² d[lambda] / [oo] ab² d[lambda] A0 = | ------------------------------------------------------- = | -----------------------------------, (13) _/ 0 (a² + [lambda])[root][4(a² + [lambda])(b² + [lambda])²] _/ 0 2(a² + [lambda])^3/2 (b² + [lambda])
A0 + 2B0 = 1, (14)
A0 B0 1 - A0 1 [alpha] = ------, [beta] = ------ = ------ = ------------. (15) 1 - A0 1 - B0 1 + A0 1 + 2[alpha]
The length of the shot being denoted by l and the calibre by d, and the length in calibres by x
l/d = 2a/2b = x, (16)
x 1 A0 = ----------- ch^(-1)x - ------, (17) (x²- 1)^3/2 x² - 1
-x x² 2B0 = ----------- ch^(-1)x + ------, (18) x² - 1)^3/2 x² + 1
x sh^(-1) [root](x²-1) x x²A0 + 2B0 = ---------------------- = ------------ log [x + [root](x² - 1)]. (19) [root](x²-1) [root](x²-1)
If [sigma] denotes the density of the metal, and if the shell has a cavity homothetic with the external ellipsoidal shape, a fraction f of the linear scale; then the volume of a round shot being 1/6 [pi] d^3, and 1/6 [pi] d^3 x of a shot x calibres long
W = 1/6 [pi] d^3 x (i - f^3) [sigma], (20)
d² Wk1² = 1/6 [pi] d^3 x -- (1 - f^5) [sigma], (21) 10
l² + d² Wk2² = 1/6 [pi] d^3 x ------- (1 - f^5) [sigma]. (22) 20
If [rho] denotes the density of the air or medium
W´ = 1/6 [pi] d^3 x [rho], (23)
W´ 1 [rho] --- = ------ -------, (24) W 1 - f³ [sigma]
k1² 1 1 - f^5 k2² x² + 1 --- = --- -------, ---- = ------, (25) d² 10 1 - f³ k1² 2
[rho] x² + 1 tan² [delta] = ------- ([beta] - [alpha])-------------, (26) [sigma] 1/5 (1 - f^5)
in which [sigma]/[rho] may be replaced by 800 times the S.G. of the metal, taking water as 800 times denser than air on the average, in round numbers, and formula (10) may be written n tan [delta] = [pi], or n[delta] = 180, when [delta] is a small angle, and given in degrees.
From this formula (26) the table following has been calculated by A. G. Hadcock, and the results are in agreement with practical experience.
52. In the steady motion the centre of the shot describes a helix, with axial velocity
/ c1 \ u cos [theta]= v sin [theta] = ( l + -- tan² [theta] ) u cos [theta] [asympt] u sec [theta], (1) \ c2 /
and transverse velocity
/ c1 \ u sin [theta] - v cos [theta] = ( l - -- ) u sin [theta] [asympt] ([beta] - [alpha]) u sin [theta]; (2) \ c2 /
and the time of completing a turn of the spiral is 2[pi]/[mu].
When [mu] has the critical value in (7),
2[pi] 4[pi] C2 2[pi] ----- = ----- -- cos [theta] = ----- (x² + 1) cos [theta], (3) [mu] p C1 p
which makes the circumference of the cylinder on which the helix is wrapped
2[pi] 2[pi]u -----(u sin [theta] - v cos [theta] = ------ ([beta] - [alpha]) (x² + 1) sin² [theta] cos [theta] [mu] p
= nd ([beta] - [alpha]) (x² + 1) sin [theta] cos [theta], (4)
and the length of one turn of the helix
2[pi] ----- (u cos [theta] + v sin [theta] ) = nd(x² + 1); (5) [mu]
thus for x = 3, the length is 10 times the pitch of the rifling.
53. _The Motion of a Perforated Solid in Liquid._--In the preceding investigation, the liquid stops dead when the body is brought to rest; and when the body is in motion the surrounding liquid moves in a uniform manner with respect to axes fixed in the body, and the force experienced by the body from the pressure of the liquid on its surface is the opposite of that required to change the motion of the liquid; this has been expressed by the dynamical equations given above. But if the body is perforated, the liquid can circulate through a hole, in reentrant stream lines linked with the body, even while the body is at rest; and no reaction from the surface can influence this circulation, which may be supposed started in the ideal manner described in § 29, by the application of impulsive pressure across an ideal membrane closing the hole, by means of ideal mechanism connected with the body. The body is held fixed, and the reaction of the mechanism and the resultant of the impulsive pressure on the surface are a measure of the impulse, linear [xi], [eta], [zeta], and angular [lambda], [mu], [nu], required to start the circulation.
This impulse will remain of constant magnitude, and fixed relatively to the body, which thus experiences an additional reaction from the circulation which is the opposite of the force required to change the position in space of the circulation impulse; and these extra forces must be taken into account in the dynamical equations.
An article may be consulted in the _Phil. Mag._, April 1893, by G. H. Bryan, in which the analytical equations of motion are deduced of a perforated solid in liquid, from considerations purely hydrodynamical.
The effect of an external circulation of vortex motion on the motion of a cylinder has been investigated in § 29; a similar procedure will show the influence of circulation through a hole in a solid, taking as the simplest illustration a ring-shaped figure, with uniplanar motion, and denoting by [xi] the resultant axial linear momentum of the circulation.
As the ring is moved from O to O´ in time t, with velocity Q, and angular velocity R, the components of liquid momentum change from
[alpha]M´U + [xi] and [beta]M´V along Ox and Oy
to
[alpha]M´U´ + [xi] and [beta]M´V´ along O´x´ and O´y´, (1)
the axis of the ring changing from Ox to O´x´; and
U = Q cos [theta], V = Q sin [theta],
U´ = Q cos ([theta] - Rt), V´ = Q sin ([theta] - Rt), (2)
so that the increase of the components of momentum, X1, Y1, and N1, linear and angular, are
X1 = ([alpha]M´U´ + [xi])[cos] Rt - [alpha]M´U - [xi] - [beta]M´V´ sin Rt = ([alpha] - [beta])M´Q sin ([theta] - Rt) sin Rt - [xi] ver Rt (3)
Y1 = ([alpha]M´U´ + [xi]) sin Rt + [beta]M´V´ cos Rt - [beta]M´V = ([alpha] - [beta])M´Q cos ([theta] - Rt) sin Rt + [xi] sin RT, (4)
N1 = [ -([alpha]M´U´ + [xi]) sin ([theta] - Rt) + [beta]M´V´ cos ([theta] - Rt)OO´ = [ -([alpha] - [beta])´Q cos ([theta] - Rt) sin ([theta] - Rt) - [xi] sin ([theta] - Rt)]Qt. (5)
The components of force, X, Y, and N, acting on the liquid at O, and reacting on the body, are then
X = lt. X1/t = ([alpha] - [beta])M´QR sin [theta] = ([alpha] - [beta])M´VR, (6)
Y = lt. Y1/t = ([alpha] - [beta])M´QR cos [theta] + [xi]R = ([alpha] - [beta])M´UR + [xi]R, (7)
Z = lt. Z1/t = -([alpha] - [beta])M´Q² sin [theta]cos[theta] - [xi]Q sin [theta] = [-([alpha] - [beta])M´U + [xi]]V. (8)
Now suppose the cylinder is free; the additional forces acting on the body are the components of kinetic reaction of the liquid
/ dU \ / dV \ dR -[alpha]M´ ( -- - VR ), -[beta]M´ ( -- + UR ), [epsilon]C´--, (9) \ dt / \ dt / dt
so that its equations of motion are
/ dU \ / dU \ M ( -- -VR ) = -[alpha]M´ ( -- - VR ) - ([alpha] - [beta])M´VR, (10) \ dt / \ dt /
/ dV \ / dV \ M ( -- + UR ) = -[beta]M´ ( -- + UR ) - ([alpha] - [beta])M´UR - [xi]R, (11) \ dt / \ dt /
dR dR C-- = -[epsilon]C´-- + ([alpha] - [beta])M´UV + [xi]V; (12) dt dt
and putting as before
M + [alpha]M´ = c1, M + [beta]M´ = c2, C + [epsilon]C´ = C3, (13)
dU c1-- - c2 VR = 0, (14) dt
dV c2-- + (c1 U + [xi])R = 0, (15) dt
dR c3-- - (c1U + [xi] - c2 U)V = 0; (16) dt
showing the modification of the equations of plane motion, due to the component [xi] of the circulation.
The integral of (14) and (15) may be written
c1U + [xi] = F cos [theta], c2V = - F sin [theta], (17)
dx F cos² [theta] F sin² [theta] [xi] -- = U cos [theta] - V sin [theta] = -------------- + -------------- - ---- cos [theta], (18) dt c1 c2 c1
d[mu] / F F \ [xi] ----- = U sin [theta] + V cos [theta] = ( -- - -- ) sin [theta] cos [theta] - ---- sin [theta], (19) dt \ c1 c2 / c1
d²[theta] / F² F² \ F[xi] d[mu] C3--------- = ( -- - -- ) sin [theta] cos [theta] - ----- sin [theta] = F-----, (20) dt² \ c1 c2 / c1 dt _ _ d[theta] / | F² cos² [theta] F² sin² [theta] F[xi] | C3-------- = Fy = / | - --------------- - --------------- + 2----- cos [theta] + H |; (21) dt \/ |_ c1 c2 c1 _|
so that cos [theta] and y is an elliptic function of the time.
When [xi] is absent, dx/dt is always positive, and the centre of the body cannot describe loops; but with [xi], the influence may be great enough to make dx/dt change sign, and so loops occur, as shown in A. B. Basset's _Hydrodynamics_, i. 192, resembling the trochoidal curves, which can be looped, investigated in § 29 for the motion of a cylinder under gravity, when surrounded by a vortex.
The branch of hydrodynamics which discusses wave motion in a liquid or gas is given now in the articles SOUND and WAVE; while the influence of viscosity is considered under HYDRAULICS.
REFERENCES.--For the history and references to the original memoirs see _Report to the British Association_, by G. G. Stokes (1846), and W. M. Hicks (1882). See also the _Fortschritte der Mathematik_, and A. E. H. Love, "Hydrodynamik" in the _Encyklöpadie der mathematischen Wissenschaften_ (1901). (A. G. G.)
HYDROMEDUSAE, a group of marine animals, recognized as belonging to the Hydrozoa (q.v.) by the following characters. (1) The polyp (hydropolyp) is of simple structure, typically much longer than broad, without ectodermal oesophagus or mesenteries, such as are seen in the anthopolyp (see article ANTHOZOA); the mouth is usually raised above the peristome on a short conical elevation or hypostome; the ectoderm is without cilia. (2) With very few exceptions, the polyp is not the only type of individual that occurs, but alternates in the life-cycle of a given species, with a distinct type, the medusa (q.v.), while in other cases the polyp-stage may be absent altogether, so that only medusa-individuals occur in the life-cycle.
The Hydromedusae represent, therefore, a sub-class of the Hydrozoa. The only other sub-class is the Scyphomedusae (q.v.). The Hydromedusae contrast with the Scyphomedusae in the following points. (1) The polyp, when present, is without the strongly developed longitudinal retractor muscles, forming ridges (_taeniolae_) projecting into the digestive cavity, seen in the scyphistoma or scyphopolyp. (2) The medusa, when present, has a velum and is hence said to be _craspedote_; the nervous system forms two continuous rings running above and below the velum; the margin of the umbrella is not lobed (except in Narcomedusae) but entire; there are characteristic differences in the sense-organs (see below, and SCYPHOMEDUSAE); and gastral filaments (phacellae), subgenital pits, &c., are absent. (3) The gonads, whether formed in the polyp or the medusa, are developed in the ectoderm.
The Hydromedusae form a widespread, dominant and highly differentiated group of animals, typically marine, and found in all seas and in all zones of marine life. Fresh-water forms, however, are also known, very few as regards species or genera, but often extremely abundant as individuals. In the British fresh-water fauna only two genera, _Hydra_ and _Cordylophora_, are found; in America occurs an additional genus, _Microhydra_. The paucity of fresh-water forms contrasts sharply, with the great abundance of marine genera common in all seas and on every shore. The species of _Hydra_, however, are extremely common and familiar inhabitants of ponds and ditches.
In fresh-water Hydromedusae the life-cycle is usually secondarily simplified, but in marine forms the life-cycle may be extremely complicated, and a given species often passes in the course of its history through widely different forms adapted to different habitats and modes of life. Apart from larval or embryonic forms there are found typically two types of person, as already stated, the polyp and the medusa, each of which may vary independently of the other, since their environment and life-conditions are usually quite different. Hence both polyp and medusa present characters for classification, and a given species, genus or other taxonomic category may be defined by polyp-characters or medusa-characters or by both combined. If our knowledge of the life-histories of these organisms were perfect, their polymorphism would present no difficulties to classification; but unfortunately this is far from being the case. In the majority of cases we do not know the polyp corresponding to a given medusa, or the medusa that arises from a given polyp.[1] Even when a medusa is seen to be budded, from a polyp under observation in an aquarium, the difficulty is not always solved, since the freshly-liberated, immature medusa may differ greatly from the full-grown, sexually-mature medusa after several months of life on the high seas (see figs. 11, B, C, and 59, a, b, c). To establish the exact relationship it is necessary not only to breed but to rear the medusa, which cannot always be done in confinement. The alternative is to fish all stages of the medusa in its growth in the open sea, a slow and laborious method in which the chance of error is very great, unless the series of stages is very complete.
At present, therefore, classifications of the Hydromedusae have a more or less tentative character, and are liable to revision with increased knowledge of the life-histories of these organisms. Many groups bear at present two names, the one representing the group as defined by polyp-characters, the other as defined by medusa-characters. It is not even possible in all cases to be certain that the polyp-group corresponds exactly to the medusa-group, especially in minor systematic categories, such as families.
The following is the main outline of the classification that is Adopted in the present article. Groups founded on polyp-characters are printed in ordinary type, those founded on medusa-characters in italics. For definitions of the groups see below.
Sub-class Hydromedusae (_Hydrozoa Craspedota_).
Order I. Eleutheroblastea. " II. Hydroidea (_Leptolinae_). Sub-order 1. Gymnoblastea (_Anthomedusae_). " 2. Calyptoblastea (_Leptomedusae_). Order III. Hydrocorallinae. " IV. Graptolitoidea. " V. Trachylinae. Sub-order 1. _Trachomedusae_. " 2. _Narcomedusae_. Order VI. Siphonophora. Sub-order 1. Chondrophorida. " 2. Calycophorida. " 3. Physophorida. " 4. Cystophorida.
_Organization and Morphology of the Hydromedusae._
As already stated, there occur in the Hydromedusae two distinct types of person, the polyp and the medusa; and either of them is capable of non-sexual reproduction by budding, a process which may lead to the formation of colonies, composed of more or fewer individuals combined and connected together. The morphology of the group thus falls naturally into four sections--(1) the hydropolyp, (2) the polyp-colony, (3) the hydromedusa, (4) the medusa-colonies. Since, however, medusa-colonies occur only in one group, the Siphonophora, and divergent views are held with regard to the morphological interpretation of the members of a siphonophore, only the first three of the above subdivisions of hydromedusa morphology will be dealt with here in a general way, and the morphology of the Siphonophora will be considered under the heading of the group itself.
1. _The Hydropolyp_ (fig. 1)--The general characters of this organism are described above and in the articles HYDROZOA and POLYP. It is rarely free, but usually fixed and incapable of locomotion. The foot by which it is attached often sends out root-like processes--the _hydrorhiza_ (c). The column (b) is generally long, slender and stalk-like (_hydrocaulus_). Just below the crown of tentacles, however, the body widens out to form a "head," termed, the _hydranth_ (a), containing a stomach-like dilatation of the digestive cavity. On the upper face of the hydranth the crown of tentacles (t) surrounds the peristome, from which rises the conical hypostome, bearing the mouth at its extremity. The general ectoderm covering the surface of the body has entirely lost the cilia present in the earlier larval stages (planula), and may be naked, or clothed in a cuticle or exoskeleton, the perisarc (ps), which in its simplest condition is a chitinous membrane secreted by the ectoderm. The perisarc when present invests the hydrorhiza and hydrocaulus; it may stop short below the hydranth, or it may extend farther. In general there are two types of exoskeleton, characteristic of the two principal divisions of the Hydroidea. In the Gymnoblastea the perisarc either stops below the hydranth, or, if continued on to it, forms a closely-fitting investment extending as a thin cuticle as far as the bases of the tentacles (e.g. _Bimeria_, see G. J. Allman [1],[2] pl. xii. figs, 1 and 3). In the Calyptoblastea the perisarc is always continued above the hydrocaulus, and forms a cup, the hydrangium or hydrotheca (h, t), standing off from the body, into which the hydranth can be retracted for shelter and protection.
The architecture of the hydropolyp, simple though it be, furnishes a long series of variations affecting each part of the body. The greatest variation, however, is seen in the tentacles. As regards number, we find in the aberrant forms _Protohydra_ and _Microhydra_ tentacles entirely absent. In the curious hydroid _Monobrachium_ a single tentacle is present, and the same is the case in _Clathrozoon_; in _Amphibrachium_ and in _Lar_ (fig. 11, A) the polyp bears two tentacles only. The reduction of the tentacles in all these forms may be correlated with their mode of life, and especially with living in a constant current of water, which brings food-particles always from one direction and renders a complete whorl or circle of tentacles unnecessary. Thus _Microhydra_ lives amongst Bryozoa, and appears to utilize the currents produced by these animals. _Protohydra_ occurs in oyster-banks and _Monobrachium_ also grows on the shells of bivalves, and both these hydroids probably fish in the currents produced by the lamellibranchs. _Amphibrachium_ grows in the tissues of a sponge, _Euplectella_, and protrudes its hydranth into the canal-system of the sponge; and _Lar_ grows on the tubes of the worm _Sabella_. With the exception of these forms, reduced for the most part in correlation with a semi-parasitic mode of life, the tentacles are usually numerous. It is rare to find in the polyp a regular, symmetrical disposition of the tentacles as in the medusa. The primitive number of four in a whorl is seen, however, in _Stauridium_ (fig. 2) and _Cladonema_ (Allman [1], pl. xvii.), and in _Clavatella_ each whorl consists regularly of eight (Allman, _loc. cit._ pl. xviii.). As a rule, however, the number in a whorl is irregular. The tentacles may form a single whorl, or more than one; thus in _Corymorpha_ (fig. 3) and _Tubularia_ (fig. 4) there are two circlets; in _Stauridium_ (fig. 2) several; in _Coryne_ and _Cordylophora_ the tentacles are scattered irregularly over the elongated hydranth.
As regards form, the tentacles show a number of types, of which the most important are (1) filiform, _i.e._ cylindrical or tapering from base to extremity, as in _Clava_ (fig. 5); (2) capitate, i.e. knobbed at the extremity, as in _Coryne_ (see Allman, loc. cit. pl. iv.); (3) branched, a rare form in the polyp, but seen in _Cladocoryne_ (see Allman, loc. cit. p. 380, fig. 82). Sometimes more than one type of form is found in the same polyp; in _Pennaria_ and _Stauridium_ (fig. 2) the upper whorls are capitate, the lower filiform. Finally, as regards structure, the tentacles may retain their primitive hollow nature, or become solid by obliteration of the axial cavity.
The hypostome of the hydropolyp may be small, or, on the other hand, as in _Eudendrium_ (Allman, loc. cit. pls. xiii., xiv.), large and trumpet-shaped. In the curious polyp _Myriothela_ the body of the polyp is differentiated into nutritive and reproductive portions.
_Histology._--The ectoderm of the hydropolyp is chiefly sensory, contractile and protective in function. It may also be glandular in places. It consists of two regions, an external epithelial layer and a more internal sub-epithelial layer.
The epithelial layer consists of (1) so-called "indifferent" cells secreting the perisarc or cuticle and modified to form glandular cells in places; for example, the adhesive cells in the foot. (2) Sensory cells, which may be fairly numerous in places, especially on the tentacles, but which occur always scattered and isolated, never aggregated to form sense-organs as in the medusa. (3) Contractile or myo-epithelial cells, with the cell prolonged at the base into a contractile muscle-fibre (fig. 6, B). In the hydropolyp the ectodermal muscle-fibres are always directed longitudinally. Belonging primarily to the epithelial layer, the muscular cells may become secondarily sub-epithelial.
The sub-epithelial layer consists primarily of the so-called interstitial cells, lodged between the narrowed basal portions of the epithelial cells. From them are developed two distinct types of histological elements; the genital cells and the cnidoblasts or mother-cells of the nematocysts. The sub-epithelial layer thus primarily constituted may be recruited by immigration from without of other elements, more especially by nervous (ganglion) cells and muscle-cells derived from the epithelial layer. In its fullest development, therefore, the sub-epithelial layer consists of four classes of cell-elements.
The genital cells are simple wandering cells (archaeocytes), at first minute and without any specially distinctive features, until they begin to develop into germ-cells. According to Wulfert [60] the primitive germ-cells of _Gonothyraea_ can be distinguished soon after the fixation of the planula, appearing amongst the interstitial cells of the ectoderm. The germ-cells are capable of extensive migrations, not only in the body of the same polyp, but also from parent to bud through many non-sexual generations of polyps in a colony (A. Weismann [58]).
The cnidoblasts are the mother-cells of the nematocysts, each cell producing one nematocyst in its interior. The complete nematocyst (fig. 7) is a spherical or oval capsule containing a hollow thread, usually barbed, coiled in its interior. The capsule has a double wall, an outer one (o.c.), tough and rigid in nature, and an inner one (i.c.) of more flexible consistence. The outer wall of the capsule is incomplete at one pole, leaving an aperture through which the thread is discharged. The inner membrane is continuous with the wall of the hollow thread at a spot immediately below the aperture in the outer wall, so that the thread itself (f) is simply a hollow prolongation of the wall of the inner capsule inverted and pushed into its cavity. The entire nematocyst is enclosed in the cnidoblast which formed it. When the nematocyst is completely developed, the cnidoblast passes outwards so as to occupy a superficial position in the ectoderm, and a delicate protoplasmic process of sensory nature, termed the _cnidocil_ (cn) projects from the cnidoblast like a fine hair or cilium. Many points in the development and mechanism of the nematocyst are disputed, but it is tolerably certain (1) that the cnidocil is of sensory nature, and that stimulation, by contact with prey or in other ways, causes a reflex discharge of the nematocyst; (2) that the discharge is an explosive change whereby the in-turned thread is suddenly everted and turned inside out, being thus shot through the opening in the outer wall of the capsule, and forced violently into the tissues of the prey, or, it may be, of an enemy; (3) that the thread inflicts not merely a mechanical wound, but instils an irritant poison, numbing and paralysing in its action. The points most in dispute are, first, how the explosive discharge is brought about, whether by pressure exerted external to the capsule (i.e. by contraction of the cnidoblast) or by internal pressure. N. Iwanzov [27] has brought forward strong grounds for the latter view, pointing out that the cnidoblast has no contractile mechanism and that measurements show discharged capsules to be on the average slightly larger than undischarged ones. He believes that the capsule contains a substance which swells very rapidly when brought into contact with water, and that in the undischarged condition the capsule has its opening closed by a plug of protoplasm (x, fig. 7) which prevents access of water to the contents; when the cnidocil is stimulated it sets in action a mechanism or perhaps a series of chemical changes by which the plug is dissolved or removed; as a result water penetrates into the capsule and causes its contents to swell, with the result that the thread is everted violently. A second point of dispute concerns the spot at which the poison is lodged. Iwanzov believes it to be contained within the thread itself before discharge, and to be introduced into the tissues of the prey by the eversion of the thread. A third point of dispute is whether the nematocysts are formed _in situ_, or whether the cnidoblasts migrate with them to the region where they are most needed; the fact that in _Hydra_, for example, there are no interstitial cells in the tentacles, where nematocysts are very abundant, is certainly in favour of the view that the cnidoblasts migrate on to the tentacles from the body, and that like the genital cells the cnidoblasts are wandering cells.
The muscular tissue consists primarily of processes from the bases of the epithelial cells, processes which are contractile in nature and may be distinctly striated. A further stage in evolution is that the muscle-cells lose their connexion with the epithelium and come to lie entirely beneath it, forming a sub-epithelial contractile layer, developed chiefly in the tentacles of the polyp. The evolution of the ganglion-cells, is probably similar; an epithelial cell develops processes of nervous nature from the base, which come into connexion with the bases of the sensory cells, with the muscular cells, and with the similar processes of other nerve-cells; next the nerve-cell loses its connexion with the outer epithelium and becomes a sub-epithelial ganglion-cell which is closely connected with the muscular layer, conveying stimuli from the sensory cells to the contractile elements. The ganglion-cells of Hydromedusae are generally very small. In the polyp the nervous tissue is always in the form of a scattered plexus, never concentrated to form a definite nervous system as in the medusa.
The endoderm of the polyp is typically a flagellated epithelium of large cells (fig. 6), from the bases of which arise contractile muscular processes lying in the plane of the transverse section of the body. In different parts of the coelenteron the endoderm may be of three principal types--(1) digestive endoderm, the primitive type, with cells of large size and considerably vacuolated, found in the hydranth; some of these cells may become special glandular cells, without flagella or contractile processes; (2) circulatory endoderm, without vacuoles and without basal contractile processes, found in the hydrorhiza and hydrocaulus; (3) supporting endoderm (fig. 8), seen in solid tentacles as a row of cubical vacuolated cells, occupying the axis of the tentacle, greatly resembling notochordal tissue, particularly that of _Amphioxus_ at a certain stage of development; as a fourth variety of endodermal cells excretory cells should perhaps be reckoned, as seen in the pores in the foot of _Hydra_ and elsewhere (cf. C. Chun, HYDROZOA [1], pp. 314, 315).
The mesogloea in the hydropolyp is a thin elastic layer, in which may be lodged the muscular fibres and ganglion cells mentioned above, but which never contains any connective tissue or skeletogenous cells or any other kind of special mesogloeal corpuscles.
2. _The Polyp-colony._--All known hydropolyps possess the power of reproduction by budding, and the buds produced may become either polyps or medusae. The buds may all become detached after a time and give rise to separate and independent individuals, as in the common _Hydra_, in which only polyp-individuals are produced and sexual elements are developed upon the polyps themselves; or, on the other hand, the polyp-individuals produced by budding may remain permanently in connexion with the parent polyp, in which case sexual elements are never developed on polyp-individuals but only on medusa-individuals, and a true colony is formed. Thus the typical hydroid colony starts from a "founder" polyp, which in the vast majority of cases is fixed, but which may be floating, as in _Nemopsis_, _Pelagohydra_, &c. The founder-polyp usually produces by budding polyp-individuals, and these in their turn produce other buds. The polyps are all non-sexual individuals whose function is purely nutritive. After a time the polyps, or certain of them, produce by budding medusa-individuals, which sooner or later develop sexual elements; in some cases, however, the founder-polyp remains solitary, that is to say, does not produce polyp-buds, but only medusa-buds, from the first (_Corymorpha_, fig. 3, _Myriothela_, &c.). In primitive forms the medusa-individuals are set free before reaching sexual maturity and do not contribute anything to the colony. In other cases, however, the medusa-individuals become sexually mature while still attached to the parent polyp, and are then not set free at all, but become appanages of the hydroid colony and undergo degenerative changes leading to reduction and even to complete obliteration of their original medusan structure. In this way the hydroid colony becomes composed of two portions of different function, the nutritive "trophosome," composed of non-sexual polyps, and the reproductive "gonosome," composed of sexual medusa-individuals, which never exercise a nutritive function while attached to the colony. As a general rule polyp-buds are produced from the hydrorhiza and hydrocaulus, while medusa-buds are formed on the hydranth. In some cases, however, medusa-buds are formed on the hydrorhiza, as in Hydrocorallines.
In such a colony of connected individuals, the exact limits of the separate "persons" are not always clearly marked out. Hence it is necessary to distinguish between, first, the "zooids," indicated in the case of the polyps by the hydranths, each with mouth and tentacles; and, secondly, the "coenosarc," or common flesh, which cannot be assigned more to one individual than another, but consists of a more or less complicated network of tubes, corresponding to the hydrocaulus and hydrorhiza of the primitive independent polyp-individual. The coenosarc constitutes a system by which the digestive cavity of any one polyp is put into communication with that of any other individual either of the trophosome or gonosome. In this manner the food absorbed by one individual contributes to the welfare of the whole colony, and the coenosarc has the function of circulating and distributing nutriment through the colony.
The hydroid colony shows many variations in form and architecture which depend simply upon differences in the methods in which polyps are budded.
In the first place, buds may be produced only from the hydrorhiza, which grows out and branches to form a basal _stolon_, typically net-like, spreading over the substratum to which the founder-polyp attached itself. From the stolon the daughter-polyps grow up vertically. The result is a spreading or creeping colony, with the coenosarc in the form of a root-like horizontal network (fig. 5, B; 11, A). Such a colony may undergo two principal modifications. The meshes of the basal network may become very small or virtually obliterated, so that the coenosarc becomes a crust of tubes tending to fuse together, and covered over by a common perisarc. Encrusting colonies of this kind are seen in _Clava squamata_ (fig. 5, A) and _Hydractinia_ (figs. 9, 10), the latter having the perisarc calcified. A further very important modification is seen when the tubes of the basal perisarc do not remain spread out in one plane, but grow in all planes forming a felt-work; the result is a massive colony, such as is seen in the so-called Hydrocorallines (fig. 60), where the interspaces between the coenosarcal tubes are filled up with calcareous matter, or _coenosteum_, replacing the chitinous perisarc. The result is a stony, solid mass, which contributes to the building up of coral reefs. In massive colonies of this kind no sharp distinction can be drawn between hydrorhiza and hydrocaulus in the coenosarc; it is practically all hydrorhiza. Massive colonies may assume various forms and are often branching or tree-like. A further peculiarity of this type of colony is that the entire coenosarcal complex is covered externally by a common layer of ectoderm; it is not clear how this covering layer is developed.
In the second place, the buds may be produced from the hydrocaulus, growing out laterally from it; the result is an arborescent, tree-like colony (figs. 12, 13). Budding from the hydrocaulus may be combined with budding from the hydrorhiza, so that numerous branching colonies arise from a common basal stolon. In the formation of arborescent colonies, two sharply distinct types of budding are found, which are best described in botanical terminology as the monopodial or racemose, and the sympodial or cymose types respectively; each is characteristic of one of the two sub-orders of the Hydroidea, the Gymnoblastea and Calyptoblastea.
In the monopodial method (figs. 12, 14) the founder-polyp is, theoretically, of unlimited growth in a vertical direction, and as it grows up it throws out buds right and left alternately, so that the first bud produced by it is the lowest down, the second bud is above the first, the third above this again, and so on. Each bud produced by the founder proceeds to grow and to bud in the same way as the founder did, producing a side branch of the main stem. Hence, in a colony of gymnoblastic hydroids, the oldest polyp of each system, that is to say, of the main stem or of a branch, is the topmost polyp; the youngest polyp of the system is the one nearest to the topmost polyp; and the axis of the system is a true axis.
In the sympodial method of budding, on the other hand, the founder-polyp is of limited growth, and forms a bud from its side, which is also of limited growth, and forms a bud in its turn, and so on (figs. 15, 16). Hence, in a colony of calyptoblastic hydroids, the oldest polyp of a system is the lowest; the youngest polyp is the topmost one; and the axis of the system is a false axis composed of portions of each of the consecutive polyps. In this method of budding there are two types. In one, the biserial type (fig. 15), the polyps produce buds right and left alternately, so that the hydranths are arranged in a zigzag fashion, forming a "scorpioid cyme," as in _Obelia_ and _Sertularia_. In the other, the uniserial type (fig. 16), the buds are formed always on the same side, forming a "helicoid cyme," as in _Hydrallmania_, according to H. Driesch, in which, however, the primitively uniserial arrangement becomes masked later by secondary torsions of the hydranths.
In a colony formed by sympodial budding, a polyp always produces first a bud, which contributes to the system to which it belongs, i.e. continues the stem or branch of which its parent forms a part. The polyp may then form a second bud, which becomes the starting point of a new system, the beginning, that is, of a new branch; and even a third bud, starting yet another system, may be produced from the same polyp. Hence the colonies of Calyptoblastea may be complexly branched, and the budding may be biserial throughout, uniserial throughout, or partly one, partly the other. Thus in _Plumularidae_ (figs. 17, 18) there is formed a main stem by biserial budding; each polyp on the main stem forms a second bud, which usually forms a side branch or _pinnule_ by uniserial budding. In this way are formed the familiar feathery colonies of _Plumularia_, in which the pinnules are all in one plane, while in the allied _Antennularia_ the pinnules are arranged in whorls round the main biserial stem. The pinnules never branch again, since in the uniserial mode of budding a polyp never forms a second polyp-bud. On the other hand, a polyp on the main stem may form a second bud which, instead of forming a pinnule by uniserial budding, produces by biserial budding a branch, from which pinnules arise as from the main stem (fig. 18--3, 6). Or a polyp on the main stem, after having budded a second time to form a pinnule, may give rise to a third bud, which starts a new biserial system, from which uniserial pinnules arise as from the main stem--type of _Aglaophenia_ (fig. 19). The laws of budding in hydroids have been worked out in an interesting manner by H. Driesch [13], to whose memoirs the reader must be referred for further details.
_Individualization of Polyp-Colonies._--As in other cases where animal colonies are formed by organic union of separate individuals, there is ever a tendency for the polyp-colony as a whole to act as a single individual, and for the members to become subordinated to the needs of the colony and to undergo specialization for particular functions, with the result that they simulate organs and their individuality becomes masked to a greater or less degree. Perhaps the earliest of such specializations is connected with the reproductive function. Whereas primitively any polyp in a colony may produce medusa-buds, in many hydroid colonies medusae are budded only by certain polyps termed _blastostyles_ (fig. 10, b). At first not differing in any way from other polyps (fig. 5), the blastostyles gradually lose their nutritive function and the organs connected with it; the mouth and tentacles disappear, and the blastostyle obtains the nutriment necessary for its activity by way of the coenosarc. In the Calyptoblastea, where the polyps are protected by special capsules of the perisarc, the _gonothecae_ enclosing the blastostyles differ from the hydrothecae protecting the hydranths (fig. 54).
In other colonies the two functions of the nutritive polyp, namely, capture and digestion of food, may be shared between different polyps (fig. 10). One class of polyps, the _dactylozoids_ (dz), lose their mouth and stomach, and become elongated and tentacle-like, showing great activity of movement. Another class, the _gastrozoids_ (gz), have the tentacles reduced or absent, but have the mouth and stomach enlarged. The dactylozoids capture food, and pass it on to the gastrozoids, which swallow and digest it.
Besides the three types of individual above mentioned, there are other appendages of hydroid colonies, of which the individuality is doubtful. Such are the "guard-polyps" (machopolyps) of _Plumularidae_, which are often regarded as individuals of the nature of dactylozoids, but from a study of the mode of budding in this hydroid family Driesch concluded that the guard-polyps were not true polyp-individuals, although each is enclosed in a small protecting cup of the perisarc, known as a nematophore. Again, the spines arising from the basal crust of _Podocoryne_ have been interpreted by some authors as reduced polyps.
3. _The Medusa._--In the Hydromedusae the medusa-individual occurs, as already stated, in one of two conditions, either as an independent organism leading a true life in the open seas, or as a subordinate individuality in the hydroid colony, from which it is never set free; it then becomes a mere reproductive appendage or _gonophore_, losing successively its organs of sense, locomotion and nutrition, until its medusoid nature and organization become scarcely recognizable. Hence it is convenient to consider the morphology of the medusa from these two aspects.
(a) _The Medusa as an Independent Organism._--The general structure and characteristics of the medusa are described elsewhere (see articles HYDROZOA and MEDUSA), and it is only necessary here to deal with the peculiarities of the Hydromedusa.
As regards habit of life the vast majority of Hydromedusae are pelagic organisms, floating on the surface of the open sea, propelling themselves feebly by the pumping movements of the umbrella produced by contraction of the sub-umbral musculature, and capturing their prey with their tentacles. The genera _Cladonema_ (fig. 20) and _Clavatella_ (fig. 21), however, are ambulatory, creeping forms, living in rock-pools and walking, as it were, on the tips of the proximal branches of each of the tentacles, while the remaining branches serve for capture of food. _Cladonema_ still has the typical medusan structure, and is able to swim about, but in _Clavatella_ the umbrella is so much reduced, that swimming is no longer possible. The remarkable medusa _Mnestra parasites_ is ecto-parasitic throughout life on the pelagic mollusc _Phyllirrhoe_, attached to it by the sub-umbral surface, and its tentacles have become rudimentary or absent. It is interesting to note that _Mnestra_ has been shown by J. W. Fewkes [15] and R. T. Günther [19] to belong to the same family (_Cladonemidae_) as _Cladonema_ and _Clavatella_, and it is reasonable to suppose that the non-parasitic ancestor of _Mnestra_ was, like the other two genera, an ambulatory medusa which acquired louse-like habits. In some species of the genus _Cunina_ (Narcomedusae) the youngest individuals (actinulae) are parasitic on other medusae (see below), but in later life the parasitic habit is abandoned. No other instances are known of sessile habit in Hydromedusae.
The external form of the Hydromedusae varies from that of a deep bell or thimble, characteristic of the Anthomedusae, to the shallow saucer-like form characteristic of the Leptomedusae. It is usual for the umbrella to have an even, circular, uninterrupted margin; but in the order Narcomedusae secondary down-growths between the tentacles produce a lobed, indented margin to the umbrella. The marginal tentacles are rarely absent in non-parasitic forms, and are typically four in number, corresponding to the four perradii marked by the radial canals. Interradial tentacles may be also developed, so that the total number present may be increased to eight or to an indefinitely large number. In _Willia_, _Geryonia_, &c., however, the tentacles and radial canals are on the plan of six instead of four (figs. 11 and 26). On the other hand, in some cases the tentacles are less in number than the perradii; in _Corymorpha_ (figs. 3 and 22) there is but a single tentacle, while two are found in _Amphinema_ and _Gemmaria_ (Anthomedusae), and in _Solmundella bitentaculata_ (fig. 67) and _Aeginopsis hensenii_ (fig. 23) (Narcomedusae). The tentacles also vary considerably in other ways than in number: first, in form, being usually simple, with a basal bulb, but in _Cladonemidae_ they are branched, often in complicated fashion; secondly, in grouping, being usually given off singly, and at regular intervals from the margin of the umbrella, but in _Margelidae_ and in some Trachomedusae they are given off in tufts or bunches (fig. 24); thirdly, in position and origin, being usually implanted on the extreme edge of the umbrella, but in Narcomedusae they become secondarily shifted and are given off high up on the ex-umbrella (figs. 23 and 25); and, fourthly, in structure, being hollow or solid, as in the polyp. In some medusae, for instance, the remarkable deep-sea family _Pectyllidae_, the tentacles may bear suckers, by which the animal may attach itself temporarily. It should be mentioned finally that the tentacles are very contractile and extensible, and may therefore present themselves, in one and the same individual, as long, drawn-out threads, or in the form of short corkscrew-like ringlets; they may stream downwards from the sub-umbrella, or be held out horizontally, or be directed upwards over the ex-umbrella (fig. 23). Each species of medusa usually has a characteristic method of carrying its tentacles.
The sub-umbrella invariably shows a velum as an inwardly projecting ridge or rim at its margin, within the circle of tentacles; hence the medusae of this sub-class are termed craspedote. The manubrium is absent altogether in the fresh-water medusa _Limnocnida_, in which the diameter of the mouth exceeds half that of the umbrella; on the other hand, the manubrium may attain a great length, owing to the centre of the sub-umbrella with the stomach being drawn into it, as it were, to form a long proboscis, as in _Geryonia_. The mouth may be a simple, circular pore at the extremity of the manubrium, or by folding of the edges it may become square or shaped like a Maltese cross, with four corners and four lips. The corners of the mouth may then be drawn out into lobes or lappets, which may have a branched or fringed outline (fig. 27), and in _Margelidae_ the subdivisions of the fringe simulate tentacles (fig. 24).
The internal anatomy of the Hydromedusae shows numerous variations. The stomach may be altogether lodged in the manubrium, from which the radial canals then take origin directly as in _Geryonia_ (Trachomedusae); it may be with or without gastric pouches. The radial canals may be simple or branched, primarily four, rarely six in number. The ring-canal is drawn out in Narcomedusae into festoons corresponding with the lobes of the margin, and may be obliterated altogether (_Solmaris_). In this order the radial canals are represented only by wide gastric pouches, and in the family Solmaridae are suppressed altogether, so that the tentacles and the festoons of the ring-canal arise directly from the stomach. In _Geryonia_, centripetal canals, ending blindly, arise from the ring-canal and run in a radial direction towards the centre of the umbrella (fig. 26).
_Histology of the Hydromedusa._--The histology described above for the polyp may be taken as the primitive type, from which that of the medusa differs only in greater elaboration and differentiation of the cell-elements, which are also more concentrated to form distinct tissues.
The ectoderm furnishes the general epithelial covering of the body, and the muscular tissue, nervous system and sense-organs. The external epithelium is flat on the ex-umbral surface, more columnar on the sub-umbral surface, where it forms the muscular tissue of the sub-umbrella and the velum. The nematocysts of the ectoderm may be grouped to form batteries on the tentacles, umbrellar margin and oral lappets. In places the nematocysts may be crowded so thickly as to form a tough, supporting, "chondral" tissue, resembling cartilage, chiefly developed at the margin of the umbrella and forming streaks or bars supporting the tentacles ("Tentakelspangen," _peronia_) or the tentaculocysts ("Gehörspangen," _otoporpae_).
The muscular tissue of the Hydromedusae is entirely ectodermal. The muscle-fibres arise as processes from the bases of the epithelial cells; such cells may individually become sub-epithelial in position, as in the polyp; or, in places where muscular tissue is greatly developed, as in the velum or sub-umbrella, the entire muscular epithelium may be thrown into folds in order to increase its surface, so that a deeper sub-epithelial muscular layer becomes separated completely from a more superficial body-epithelium.
In its arrangement the muscular tissue forms two systems: the one composed of striated fibres arranged circularly, that is to say, concentrically round the central axis of the umbrella; the other of non-striated fibres running longitudinally, that is to say, in a radial direction from, or (in the manubrium) parallel to, the same ideal axis. The circular system is developed continuously over the entire sub-umbral surface, and the velum represents a special local development of this system, at a region where it is able to act at the greatest mechanical advantage in producing the contractions of the umbrella by which the animal progresses. The longitudinal system is discontinuous, and is subdivided into proximal, medial and distal portions. The proximal portion forms the retractor muscles of the manubrium, or proboscis, well developed, for example, in _Geryonia_. The medial portion forms radiating tracts of fibres, the so-called "bell-muscles" running underneath, and parallel to, the radial canals; when greatly developed, as in _Tiaridae_, they form ridges, so-called mesenteries, projecting into the sub-umbral cavity. The distal portions form the muscles of the tentacles. In contrast with the polyp, the longitudinal muscle-system is entirely ectodermal, there being no endodermal muscles in craspedote medusae.
The nervous system of the medusa consists of sub-epithelial ganglion-cells, which form, in the first place, a diffuse plexus of nervous tissue, as in the polyp, but developed chiefly on the sub-umbral surface; and which are concentrated, in the second place, to form a definite central nervous system, never found in the polyp. In Hydromedusae the central nervous system forms two concentric nerve-rings at the margin of the umbrella, near the base of the velum. One, the "upper" or ex-umbral nerve-ring, is derived from the ectoderm on the ex-umbral side of the velum; it is the larger of the two rings, containing more numerous but smaller ganglion-cells, and innervates the tentacles. The other, the "lower" or sub-umbral nerve-ring, is derived from the ectoderm on the sub-umbral side of the velum; it contains fewer but larger ganglion-cells and innervates the muscles of the velum (see diagram in article MEDUSAE). The two nerve-rings are connected by fibres passing from one to the other.
The sensory cells are slender epithelial cells, often with a cilium or stiff protoplasmic process, and should perhaps be regarded as the only ectoderm-cells which retain the primitive ciliation of the larval ectoderm, otherwise lost in all Hydrozoa. The sense-cells form, in the first place, a diffuse system of scattered sensory cells, as in the polyp, developed chiefly on the manubrium, the tentacles and the margin of the umbrella, where they form a sensory ciliated epithelium covering the nerve-centres; in the second place, the sense-cells are concentrated to form definite sense-organs, situated always at the margin of the umbrella, hence often termed "marginal bodies." The possession of definite sense-organs at once distinguishes the medusa from the polyp, in which they are never found.
The sense-organs of medusae are of two kinds--first, organs sensitive to light, usually termed _ocelli_ (fig. 29); secondly, organs commonly termed _otocysts_, on account of their resemblance to the auditory vesicles of higher animals, but serving for the sense of balance and orientation, and therefore given the special name of _statocysts_ (fig. 30). The sense-organs may be _tentaculocysts_, i.e. modifications of a tentacle, as in Trachylinae, or developed from the margin of the umbrella, in no connexion with a tentacle (or, if so connected, not producing any modification in the tentacle), as in Leptolinae. In Hydromedusae the sense-organs are always exposed at the umbrellar margin (hence _Gymnophthalmata_), while in Scyphomedusae they are covered over by flaps of the umbrellar margin (hence _Steganophthalmata_).
The _statocysts_ present in general the structure of either a knob or a closed vesicle, composed of (1) indifferent supporting epithelium: (2) sensory, so-called auditory epithelium of slender cells, each bearing at its free upper end a stiff bristle and running out at its base into a nerve-fibre; (3) concrement-cells, which produce intercellular concretions, so-called otoliths. By means of vibrations or shocks transmitted through the water, or by displacements in the balance or position of the animal, the otoliths are caused to impinge against the bristles of the sensory cells, now on one side, now on the other, causing shocks or stimuli which are transmitted by the basal nerve-fibre to the central nervous system. Two stages in the development of the otocyst can be recognized, the first that of an open pit on a freely-projecting knob, in which the otoliths are exposed, the second that of a closed vesicle, in which the otoliths are covered over. Further, two distinct types of otocyst can be recognized in the Hydromedusae: that of the Leptolinae, in which the entire organ is ectodermal, concrement-cells and all, and the organ is not a tentaculocyst; and that of the Trachylinae, in which the organ is a tentaculocyst, and the concrement-cells are endodermal, derived from the endoderm of the modified tentacle, while the rest of the organ is ectodermal.
In the Leptolinae the otocysts are seen in their first stage in _Mitrocoma annae_ (fig. 31) and _Tiaropsis_ (figs. 29, 30) as an open pit at the base of the velum, on its sub-umbral side. The pit has its opening turned towards the sub-umbral cavity, while its base or fundus forms a bulge, more or less pronounced, on the ex-umbral side of the velum. At the _fundus_ are placed the concrement-cells with their conspicuous otoliths (_con_) and the inconspicuous auditory cells, which are connected with. the sub-umbral nerve-ring. From the open condition arises the closed condition very simply by closing up of the aperture of the pit. We then find the typical otocyst of the Leptomedusae, a vesicle bulging on the ex-umbral side of the velum (figs. 32, 33). The otocysts are placed on the outer wall of the vesicle (the fundus of the original pit) or on its sides; their arrangement and number vary greatly and furnish useful characters for distinguishing genera. The sense-cells are innervated, as before, from the sub-umbral nerve-ring. The inner wall of the vesicle (region of closure) is frequently thickened to form a so-called "sense-cushion," apparently a ganglionic offshoot from the sub-umbral nerve-ring. In many Leptomedusae the otocysts are very small, inconspicuous and embedded completely in the tissues; hence they may be easily overlooked in badly-preserved material, and perhaps are present in many cases where they have been said to have been wanting.
In the Trachylinae the simplest condition of the otocyst is a freely projecting club, a so-called _statorhabd_ (figs. 34, 35), representing a tentacle greatly reduced in size, covered with sensory ectodermal epithelium (_ect._), and containing an endodermal core (_end._), which is at first continuous with the endoderm of the ring-canal, but later becomes separated from it. In the endoderm large concretions are formed (_con._). Other sensory cells with long cilia cover a sort of cushion (_n.c._) at the base of the club; the club may be long and the cushion small, or the cushion large and the club small. The whole structure is innervated, like the tentacles, from the ex-umbral nerve-ring. An advance towards the second stage is seen in such a form as _Rhopalonema_ (fig. 36), where the ectoderm of the cushion rises up in a double fold to enclose the club in a protective covering forming a cup or vesicle, at first open distally; finally the opening closes and the closed vesicle may sink inwards and be found far removed from the surface, as in _Geryonia_ (fig. 37).
The _ocelli_ are seen in their simplest form as a pigmented patch of ectoderm, which consists of two kinds of cells--(1) pigment-cells, which are ordinary indifferent cells of the epithelium containing pigment-granules, and (2) visual cells, slender sensory epithelial cells of the usual type, which may develop visual cones or rods at their free extremity. The ocelli occur usually either on the inner or outer sides of the tentacles; if on the inner side, the tentacle is turned upwards and carried over the ex-umbrella, so as to expose the ocellus to the light; if the ocellus be on the outer side of a tentacle, two nerves run round the base of the tentacle to it. In other cases ocelli may occur between tentacles, as in _Tiaropsis_ (fig. 29).
The simple form of ocellus described in the foregoing paragraph may become folded into a pit or cup, the interior of which becomes filled with a clear gelatinous secretion forming a sort of vitreous body. The distal portion of the vitreous body may project from the cavity of the cup, forming a non-cellular lens as in _Lizzia_ (fig. 28). Beyond this simple condition the visual organs of the Hydromedusae do not advance, and are far from reaching the wonderful development of the eyes of Scyphomedusae (_Charybdaea_).
Besides the ordinary type of ocellus just described, there is found in one genus (_Tiaropsis_) a type of ocellus in which the visual elements are inverted, and have their cones turned away from the light, as in the human retina (fig. 30). In this case the pigment-cells are endodermal, forming a cup of pigment in which the visual cones are embedded. A similar ocellus is formed in _Aurelia_ among the Scyphomedusae (q.v.).
Other sense organs of Hydromedusae are the so-called _sense-clubs_ or _cordyli_ found in a few Leptomedusae, especially in those genera in which otocysts are inconspicuous or absent (fig. 39). Each cordylus is a tentacle-like structure with an endodermal axis containing an axial cavity which may be continuous with the ring-canal, or may be partially occluded. Externally the cordylus is covered, by very flattened ectoderm, and bears no otoliths or sense-cells, but the base of the club rests upon the ex-umbral nerve-ring. Brooks regards these organs as sensory, serving for the sense of balance, and representing a primitive stage of the tentaculocysts of Trachylinae; Linko, on the other hand, finding no nerve-elements connected with them, regards them as digestive (?) in function.
The sense-organs of the two fresh-water medusae _Limnocodium_ and _Limnocnida_ are peculiar and of rather doubtful nature (see E. T. Browne [10]).
FIG. 38.--Ocellus of _Lizzia koellikeri_. oc, Pigmented ectodermal cells; l, lens. (After Hertwig.)
The endoderm of the medusa shows the same general types of structure as in the polyp, described above. We can distinguish (1) digestive endoderm, in the stomach, often with special glandular elements; (2) circulatory endoderm, in the radial and ring-canals; (3) supporting endoderm in the axes of the tentacles and in the endoderm-lamella; the latter is primitively a double layer of cells, produced by concrescence of the ex-umbral and sub-umbral layers of the coelenteron, but it is usually found as a single layer of flattened cells (fig. 40); in _Geryonia_, however, it remains double, and the centripetal canals arise by parting of the two layers; (4) excretory endoderm, lining pores at the margin of the umbrella, occurring in certain Leptomedusae as so-called "marginal tubercles," opening, on the one hand, into the ring-canal and, on the other hand, to the exterior by "marginal funnels," which debouch into the sub-umbral cavity above the velum. As has been described above, the endoderm may also contribute to the sense-organs, but such contributions are always of an accessory nature, for instance, concrement-cells in the otocysts, pigment in the ocelli, and never of sensory nature, sense-cells being in all cases ectodermal.
The reproductive cells may be regarded as belonging primarily to neither ectoderm nor endoderm, though lodged in the ectoderm in all Hydromedusae. As described for the polyp, they are wandering cells capable of extensive migrations before reaching the particular spot at which they ripen. In the Hydromedusae they usually, if not invariably, ripen in the ectoderm, but in the neighbourhood of the main sources of nutriment, that is to say, not far from the stomach. Hence the gonads are found on the manubrium in Anthomedusae generally; on the base of the manubrium, or under the gastral pouches, or in both these situations (_Octorchidae_), or under the radial canals, in Trachomedusae; under the gastral pouches or radial canals, in Narcomedusae. When ripe, the germ-cells are dehisced directly to the exterior.
Hydromedusae are of separate sexes, the only known exception being _Amphogona apsteini_, one of the Trachomedusae (Browne [9]). Moreover, all the medusae budded from a given hydroid colony are either male or female, so that even the non-sexual polyp must be considered to have a latent sex. (In _Hydra_, on the other hand, the individual is usually hermaphrodite.) The medusa always reproduces itself sexually, and in some cases non-sexually also. The non-sexual reproduction takes the form of fission, budding or sporogony, the details of which are described below. Buds may be produced from the manubrium, radial canals, ring-canal, or tentacle-bases, or from an aboral stolon (Narcomedusae). In all cases only medusa-buds are produced, never polyp-buds.
The mesogloea of the medusa is largely developed and of great thickness in the umbrella. The sub-epithelial tissues, i.e. the nervous and muscular cells, are lodged in the mesogloea, but in Hydromedusae it never contains tissue-cells or mesogloeal corpuscles.
(b) _The Medusae as a Subordinate Individuality._--It has been shown above that polyps are budded only from polyps and that the medusae may be budded either from polyps or from medusae. In any case the daughter-individuals produced from the buds may be imagined as remaining attached to the parent and forming a colony of individuals in organic connexion with one another, and thus three possible cases arise. The first case gives a colony entirely composed of polyps, as in many Hydroidea. The second case gives a colony partly composed of polyp-individuals, partly of medusa-individuals, a possibility also realized in many colonies of Hydroidea. The third case gives a colony entirely composed of medusa-individuals, a possibility perhaps realized in the Siphonophora, which will be discussed in dealing with this group.
The first step towards the formation of a mixed hydroid colony is undoubtedly a hastening of the sexual maturity of the medusa-individual. Normally the medusae are liberated in quite an immature state; they swim away, feed, grow and become adult mature individuals. From the bionomical point of view, the medusa is to be considered as a means of spreading the species, supplementing the deficiencies of the sessile polyp. It may be, however, that increased reproductiveness becomes of greater importance to the species than wide diffusion; such a condition will be brought about if the medusae mature quickly and are either set free in a mature condition or remain in the shelter of the polyp-colony, protected from risks of a free life in the open sea. In this way the medusa sinks from an independent personality to an organ of the polyp-colony, becoming a so-called _medusoid gonophore_, or bearer of the reproductive organs, and losing gradually all organs necessary for an independent existence, namely those of sense, locomotion and nutrition.
In some cases both free medusae and gonophores may be produced from the same hydroid colony. This is the case in _Syncoryne mirabilis_ (Allman [1], p. 278) and in _Campanularia volubilis_; in the latter, free medusae are produced in summer, gonophores in winter (Duplessis [14]). Again in _Pennaria_, the male medusae are set free in a state of maturity, and have ocelli; the female medusae remain attached and have no sense organs.
The gonophores of different hydroids differ greatly in structure from one another, and form a series showing degeneration of the medusa-individual, which is gradually stripped, as it were, of its characteristic features of medusan organization and finally reduced to the simplest structure. A very early stage in the degeneration is well exemplified by the so-called "meconidium" of _Gonothyraea_ (fig. 41, A). Here the medusoid, attached by the centre of its ex-umbral surface, has lost its velum and sub-umbral muscles, its sense organs and mouth, though still retaining rudimentary tentacles. The gonads (g) are produced on the manubrium, which has a hollow endodermal axis, termed the spadix (sp.), in open communication with the coenosarc of the polyp-colony and serving for the nutrition of the generative cells. A very similar condition is seen in _Tubularia_ (fig. 41, B), where, however, the tentacles have quite disappeared, and the circular rim formed by the margin of the umbrella has nearly closed over the manubrium leaving only a small aperture through which the embryos emerge. The next step is illustrated by the female gonophores of _Cladocoryne_, where the radial and ring-canals have become obliterated by coalescence of their walls, so that the entire endoderm of the umbrella is in the condition of the endoderm-lamella. Next the opening of the umbrella closes up completely and disappears, so that the sub-umbral cavity forms a closed space surrounding the manubrium, on which the gonads are developed; such a condition is seen in the male gonophore of _Cladocoryne_ and in _Garveia_ (fig. 41, C), where, however, there is a further complication in the form of an adventitious envelope or ectotheca (ect.) split off from the gonophore as a protective covering, and not present in _Cladocoryne_. The sub-umbral cavity (s.c.) functions as a brood-space for the developing embryos, which are set free by rupture of the wall. It is evident that the outer envelope of the gonophore represents the ex-umbral ectoderm (ex.), and that the inner ectoderm lining the cavity represents the sub-umbral ectoderm of the free medusa. The next step is the gradual obliteration of the sub-umbral cavity (s.c.) by disappearance of which the sub-umbral ectoderm comes into contact with the ectoderm of the manubrium. Such a type is found in _Plumularia_ and also in _Agalma_ (fig. 41, D); centrally is seen the spadix (sp.), bearing the generative cells (g), and external to these (1) a layer of ectoderm representing the epithelium of the manubrium; (2) the layer of sub-umbral ectoderm; (3) the endoderm-lamella (e.l.); (4) the ex-umbral ectoderm (ex.); and (5) there may or may not be present also an ectotheca. Thus the gonads are covered over by at least four layers of epithelium, and since these are unnecessary, presenting merely obstacles to the dehiscence of the gonads, they gradually undergo reduction. The sub-umbral ectoderm and that covering the manubrium undergo concrescence to form a single layer (fig. 41, E), which finally disappears altogether, and the endoderm-lamella disappears. The gonophore is now reduced to its simplest condition, known as the _sporosac_ (fig. 41, F, G, H), and consists of the spadix bearing the gonads covered by a single layer of ectoderm (ex.), with or without the addition of an ectotheca. It cannot be too strongly emphasized, however, that the sporosac should not be compared simply with the manubrium of the medusa, as is sometimes done. The endodermal spadix (sp) of the sporosac represents the endoderm of the manubrium; the ectodermal lining of the sporosac (ex.) represents the ex-umbral ectoderm of the medusa; and the intervening layers, together with the sub-umbral cavity, have disappeared. The spadix, as the organ of nutrition for the gonads, may be developed in various ways, being simple (fig. 41, F) or branched (fig. 41, H); in _Eudendrium_ (fig. 41, G) it curls round the single large ovum.
The hydroid _Dicoryne_ is remarkable for the possession of gonophores, which are ciliate and become detached and swim away by means of their cilia. Each such sporosac has two long tentacle-like processes thickly ciliated.
It has been maintained that the gonads of _Hydra_ represent sporosacs or gonophores greatly reduced, with the last traces of medusoid structure completely obliterated. There is, however, no evidence whatever for this, the gonads of _Hydra_ being purely ectodermal structures, while all medusoid gonophores have an endodermal portion. _Hydra_ is, moreover, bisexual, in contrast with what is known of hydroid colonies.
In some Leptomedusae the gonads are formed on the radial canals and form protruding masses resembling sporosacs superficially, but not in structure. Allman, however, regarded this type of gonad as equivalent to a sporosac, and considered the medusa bearing them as a non-sexual organism, a "blastocheme" as he termed it, producing by budding medusoid gonophores. As medusae are known to bud medusae from the radial canals there is nothing impossible in Allman's theory, but it cannot be said to have received satisfactory proof.
_Reproduction and Ontogeny of the Hydromedusae._
Nearly every possible method of reproduction occurs amongst the Hydromedusae. In classifying methods of generation it is usual to make use of the sexual or non-sexual nature of the reproduction as a primary difference, but a more scientific classification is afforded by the distinction between tissue-cells (histocytes) and germinal cells, actual or potential (archaeocytes), amongst the constituent cells of the animal body. In this way we may distinguish, first, _vegetative_ reproduction, the result of discontinuous growth of the tissues and cell-layers of the body as a whole, leading to (1) _fission_, (2) _autotomy_, or (3) _vegetative budding_; secondly, _germinal_ reproduction, the result of the reproductive activity of the archaeocytes or germinal tissue. In germinal reproduction the proliferating cells may be _undifferentiated_, so-called primitive germ-cells, or they may be _differentiated_ as sexual cells, male or female, i.e. spermatozoa and ova. If the germ-cells are _undifferentiated_, the offspring may arise from many cells or from a single cell; the first type is (4) _germinal budding_, the second is (5) _sporogony_. If the germ-cells are _differentiated_, the offspring arises by _syngamy_ or sexual union of the ordinary type between an ovum and spermatozoon, so-called fertilization, of the ovum, or by _parthenogenesis_, i.e. development of an ovum without fertilization. The only one of these possible modes of reproduction not known to occur in Hydromedusae is parthenogenesis.
(1) True _fission_ or longitudinal division of an individual into two equal and similar daughter-individuals is not common but occurs in _Gastroblasta_, where it has been described in detail by Arnold Lang [30].
(2) _Autotomy_, sometimes termed transverse fission, is the name given to a process of unequal fission in which a portion of the body separates off with subsequent regeneration. In _Tubularia_ by a process of decapitation the hydranths may separate off and give rise to a separate individual, while the remainder of the body grows a new hydranth. Similarly in _Schizocladium_ portions of the hydrocaulus are cut off to form so-called "spores," which grow into new individuals (see Allman [1]).
(3) _Vegetative budding_ is almost universal in the Hydromedusae. By budding is understood the formation of a new individual from a fresh growth of undifferentiated material. It is convenient to distinguish buds that give rise to polyps from those that form medusae.
(a) _The Polyp._--The buds that form polyps are very simple in mode of formation. Four stages may be distinguished; the first is a simple outgrowth of both layers, ectoderm and endoderm, containing a prolongation of the coelenteric cavity; in the second stage the tentacles grow out as secondary diverticula from the side of the first outgrowth; in the third stage the mouth is formed as a perforation of the two layers; and, lastly, if the bud is to be separated, it becomes nipped off from the parent polyp and begins a free existence.
(b) _The Medusae._--Two types of budding must be distinguished--the direct, so-called, palingenetic type, and the _indirect_, so-called coenogenetic type.
The direct type of budding is rare, but is seen in _Cunina_ and _Millepora_. In _Cunina_ there arises, first, a simple outgrowth of both layers, as in a polyp-bud (fig. 43, A); in this the mouth is formed distally as a perforation (B); next the sides of the tube so formed bulge out laterally near the attachment to form the umbrella, while the distal undilated portion of the tube represents the manubrium (C); the umbrella now grows out into a number of lobes or lappets, and the tentacles and tentaculocysts grow out, the former in a notch between two lappets, the latter on the apex of each lappet (D, E); finally, the velum arises as a growth of the ectoderm alone, the whole bud shapes itself, so to speak, and the little medusa is separated off by rupture of the thin stalk connecting it with the parent (F). The direct method of medusa-budding only differs from the polyp-bud by its greater complexity of parts and organs.
The indirect mode of budding (figs. 44, 45) is the commonest method by which medusa-buds are formed. It is marked by the formation in the bud of a characteristic structure termed the _entocodon_ (_Knospenkern_, _Glockenkern_).
The first stage is a simple hollow outgrowth of both body-layers (fig. 44, A); at the tip of this is formed a thickening of the ectoderm, arising primitively as a hollow ingrowth (fig. 44, B), but more usually as a solid mass of ectoderm-cells (fig. 45, A). The ectodermal ingrowth is the entocodon (Gc.); it bulges into, and pushes down, the endoderm at the apex of the bud, and if solid it soon acquires a cavity (fig. 44, C, s.c.). The cavity of the entocodon increases continually in size, while the endoderm pushes up at the sides of it to form a cup with hollow walls, enclosing but not quite surrounding the entocodon, which remains in contact at its outer side with the ectoderm covering the bud (fig. 44, D, v). The next changes that take place are chiefly in the endoderm-cup (fig. 44, D, E); the cavity between the two walls of the cup becomes reduced by concrescence to form the radial canals (r.c.), ring-canal (c.c.), and endoderm-lamella (e.l., fig. 44, E), and at the same time the base of the cup is thrust upwards to form the manubrium (m), converting the cavity of the entocodon into a space which is crescentic or horse-shoe-like in section. Next tentacles (t, fig. 44, F) grow out from the ring-canal, and the double plate of ectoderm on the distal side of the entocodon becomes perforated, leaving a circular rim composed of two layers of ectoderm, the velum (v) of the medusa. Finally, a mouth is formed by breaking through at the apex of the manubrium, and the now fully-formed medusa becomes separated by rupture of the stalk of the bud and swims away.
If the bud, however, is destined to give rise not to a free medusa, but to a gonophore, the development is similar but becomes arrested at various points, according to the degree to which the gonophore is degenerate. The entocodon is usually formed, proving the medusoid nature of the bud, but in sporosacs the entocodon may be rudimentary or absent altogether. The process of budding as above described may be varied or complicated in various ways; thus a secondary, amnion-like, ectodermal covering or ectotheca (fig. 45, C, ect.) may be formed over all, as in _Garveia_, &c.; or the entocodon may remain solid and without cavity until after the formation of the manubrium, or may never acquire a cavity at all, as described above for the gonophores.
_Phylogenetic Significance of the Entocodon._--It is seen from the foregoing account of medusa-budding that the entocodon is a very important constituent of the bud, furnishing some of the most essential portions of the medusa; its cavity becomes the sub-umbral cavity, and its lining furnishes the ectodermal epithelium of the manubrium and of the sub-umbral cavity as far as the edge of the velum. Hence the entocodon represents a precocious formation of the sub-umbral surface, equivalent to the peristome of the polyp, differentiated in the bud prior to other portions of the organism which must be regarded as antecedent to it in phylogeny.
If the three principal organ-systems of the medusa, namely mouth, tentacles and umbrella, be considered in the light of phylogeny, it is evident that the manubrium bearing the mouth must be the oldest, as representing a common property of all the Coelentera, even of the gastrula embryo of all Enterozoa. Next in order come the tentacles, common to all Cnidaria. The special property of the medusa is the umbrella, distinguishing the medusa at once from other morphological types among the Coelentera. If, therefore, the formation of these three systems of organs took place according to a strictly phylogenetic sequence, we should expect them to appear in the order set forth above (fig. 46, Ia, b, c). The nearest approach to the phylogenetic sequence is seen in the budding of _Cunina_, where the manubrium and mouth appear first, but the umbrella is formed before the tentacles (fig. 46, IIa, b, c). In the indirect or coenogenetic method of budding, the first two members of the sequence exhibited by Cunina change places, and the umbrella is formed first, the manubrium next, and then the tentacles; the actual mouth-perforation being delayed to the very last (fig. 46, IVa, b, c). Hence the budding of medusae exemplifies very clearly a common phenomenon in development, a phylogenetic series of events completely dislocated in the ontogenetic time-sequence.
The entocodon is to be regarded, therefore, not as primarily an ingrowth of ectoderm, but rather as an upgrowth of both body-layers, in the form of a circular rim (IVa), representing the umbrellar margin; it is comparable to the bulging that forms the umbrella in the direct method of budding, but takes place before a manubrium is formed, and is greatly reduced in size, so as to become a little pit. By a simple modification, the open pit becomes a solid ectodermal ingrowth, just as in Teleostean fishes the hollow medullary tube, or the auditory pit of other vertebrate embryos, is formed at first as a solid cord of cells, which acquires a cavity secondarily. Moreover, the entocodon, however developed, gives rise at first to a closed cavity, representing a closing over of the umbrella, temporary in the bud destined to be a free medusa, but usually permanent in the sessile gonophore. As has been shown above, the closing up of the sub-umbral cavity is one of the earliest degenerative changes in the evolution of the gonophore, and we may regard it as the umbrellar fold taking on a protective function, either temporarily for the bud or permanently for the gonophore.
To sum up, the entocodon is a precocious formation of the umbrella, closing over to protect the organs in the umbrellar cavity. The possession of an entocodon proves the medusa-nature of the bud, and can only be explained on the theory that gonophores are degenerate medusae, and is inexplicable on the opposed view that medusae are derived from gonophores secondarily set free. In the sporosac, however, the medusa-individual has become so degenerate that even the documentary proof, so to speak, of its medusoid nature may have been destroyed, and only circumstantial evidence of its nature can be produced.
4. _Germinal Budding._--This method of budding is commonly described as budding from a single body-layer, instead of from both layers. The layer that produces the bud is invariably the ectoderm, i.e. the layer in which, in Hydromedusae, the generative cells are lodged; and in some cases the buds are produced in the exact spot in which later the gonads appear. From these facts, and from those of the sporogony, to be described below, we may regard budding to this type as taking place from the germinal epithelium rather than from ordinary ectoderm.
(a) _The Polyp._--Budding from the ectoderm alone has been described by A. Lang [29] in _Hydra_ and other polyps. The tissues of the bud become differentiated into ectoderm and endoderm, and the endoderm of the bud becomes secondarily continuous with that of the parent, but no part of the parental endoderm contributes to the building up of the daughter-polyp. Lang regarded this method of budding as universal in polyps, a notion disproved by O. Seeliger [52] who went to the opposite extreme and regarded the type of budding described by Lang as non-existent. In view, however, both of the statements and figures of Lang and of the facts to be described presently for medusae (_Margellium_), it is at least theoretically possible that both germinal and vegetative budding may occur in polyps as well as in medusae.
(b) _The Medusa._--The clearest instance of germinal budding is furnished by _Margellium (Rathkea) octopunctatum_, one of the _Margelidae_. The budding of this medusa has been worked out in detail by Chun (HYDROZOA, [1]), to whom the reader must be referred for the interesting laws of budding regulating the sequence and order of formation of the buds.
The buds of _Margellium_ are produced on the manubrium in each of the four interradii, and they arise from the ectoderm, that is to say, the germinal epithelium, which later gives rise to the gonads. The buds do not appear simultaneously but successively on each of the four sides of the manubrium, thus:
1 3 4 2
and secondary buds may be produced on the medusa-buds before the latter are set free as medusae. Each bud arises as a thickening of the epithelium, which first forms two or three layers (fig. 47, A), and becomes separated into a superficial layer, future ectoderm, surrounding a central mass, future endoderm (fig. 47, B). The ectodermal epithelium on the distal side of the bud becomes thickened, grows inwards, and forms a typical entocodon (fig. 37, D, E, F). The remaining development of the bud is just as described above for the indirect method of medusa-budding (fig. 47, G, H). When the bud is nearly complete, the body-wall of the parent immediately below it becomes perforated, placing the coelenteric cavity of the parent in secondary communication with that of the bud (H), doubtless for the better nutrition of the latter.
Especially noteworthy in the germinal budding of _Margellium_ is the formation of the entocodon, as in the vegetative budding of the indirect type.
5. _Sporogony._--This method of reproduction has been described by E. Metchnikoff in _Cunina_ and allied genera. In individuals either of the male or female sex, germ-cells which are quite undifferentiated and neutral in character, become amoeboid, and wander into the endoderm. They divide each into two sister-cells, one of which--the spore--becomes enveloped by the other. The spore-cell multiplies by division, while the enveloping cell is nutrient and protective. The spore cell gives rise to a "spore-larva," which is set free in the coelenteron and grows into a medusa. Whether sporogony occurs also in the polyp or not remains to be proved.
6. _Sexual Reproduction and Embryology._--The ovum of Hydromedusae is usually one of a large number of oögonia, and grows at the expense of its sister-cells. No regular follicle is formed, but the oöcyte absorbs nutriment from the remaining oögonia. In _Hydra_ the oöcyte is a large amoeboid cell, which sends out pseudopodia amongst the oögonia and absorbs nutriment from them. When the oöcyte is full grown, the residual oögonia die off and disintegrate.
The spermatogenesis and maturation and fertilization of the germ-cells present nothing out of the common and need not be described here. These processes have been studied in detail by A. Brauer [2] for _Hydra_.
The general course of the development is described in the article HYDROZOA. We may distinguish the following series of stages: (1) ovum; (2) cleavage, leading to formation of a blastula; (3) formation of an inner mass or parenchyma, the future endoderm, by immigration or delamination, leading to the so-called parenchymula-stage; (4) formation of an archenteric cavity, the future coelenteron, by a splitting of the internal parenchyma, and of a blastopore, the future mouth, by perforation at one pole, leading to the gastrula-stage; (5) the outgrowth of tentacles round the mouth (blastopore), leading to the actinula-stage; and (6) the actinula becomes the polyp or medusa in the manner described elsewhere (see articles HYDROZOA, POLYP and MEDUSA). This is the full, ideal development, which is always contracted or shortened to a greater or less extent. If the embryo is set free as a free-swimming, so-called planula-larva, in the blastula, parenchymula, or gastrula stage, then a free actinula stage is not found; if, on the other hand, a free actinula occurs, then there is no free planula stage.
The cleavage of the ovum follows two types, both seen in _Tubularia_ (Brauer [3]). In the first, a cleavage follows each nuclear division; in the second, the nuclei multiply by division a number of times, and then the ovum divides into as many blastomeres as there are nuclei present. The result of cleavage in all cases is a typical blastula, which when set free becomes oval and develops a flagellum to each cell, but when not set free, it remains spherical in form and has no flagella.
The germ-layer formation is always by immigration or delamination, never by invagination. When the blastula is oval and free-swimming the inner mass is formed by unipolar immigration from the hinder pole. When the blastula is spherical and not set free, the germ-layer formation is always multipolar, either by immigration or by delamination, i.e. by tangential division of the cells of the blastoderm, as in _Geryonia_, or by a mixture of immigration and delamination, as in _Hydra_, _Tubularia_, &c. The blastopore is formed as a secondary perforation at one spot, in free-swimming forms at the hinder pole. Formation of archenteron and blastopore may, however, be deferred till a later stage (actinula or after).
The actinula stage is usually suppressed or not set free, but it is seen in _Tubularia_ (fig. 48), where it is ambulatory, in _Gonionemus_ (Trachomedusae), and in _Cunina_ (Narcomedusae), where it is parasitic.
In Leptolinae the embryonic development culminates in a polyp, which is usually formed by fixation of a planula (parenchymula), rarely by fixation of an actinula. The planula may fix itself (1) by one end, and then becomes the hydrocaulus and hydranth, while the hydrorhiza grows out from the base; or (2) partly by one side and then gives rise to the hydrorhiza as well as to the other parts of the polyp; or (3) entirely by its side, and then forms a recumbent hydrorhiza from which a polyp appears to be budded as an upgrowth.
In Trachylinae the development produces always a medusa, and there is no polyp-stage. The medusa arises direct from the actinula-stage and there is no entocodon formed, as in the budding described above.
_Life-cycles of the Hydromedusae._--The life-cycle of the Leptolinae consists of an alternation of generations in which non-sexual individuals, polyps, produce by budding sexual individuals, medusae, which give rise by the sexual process to the non-sexual polyps again, so completing the cycle. Hence the alternation is of the type termed metagenesis. The Leptolinae are chiefly forms belonging to the inshore fauna. The Trachylinae, on the other hand, are above all oceanic forms, and have no polyp-stage, and hence there is typically no alternation in their life-cycle. It is commonly assumed that the Trachylinae are forms which have lost the alternation of generations possessed by them ancestrally, through secondary simplification of the life-cycle. Hence the Trachylinae are termed "hypogenetic" medusae to contrast them with the metagenetic Leptolinae. The whole question has, however, been argued at length by W. K. Brooks [4], who adduces strong evidence for a contrary view, that is to say, for regarding the direct type of development seen in Trachylinae as more primitive, and the metagenesis seen in Leptolinae as a secondary complication introduced into the life-cycle by the acquisition of _larval budding_. The polyp is regarded, on this view, as a form phylogenetically older than the medusa, in short, as nothing more than a sessile actinula. In Trachylinae the polyp-stage is passed over, and is represented only by the actinula as a transitory embryonic stage. In Leptolinae the actinula becomes the sessile polyp which has acquired the power of budding and producing individuals either of its own or of a higher rank; it represents a persistent larval stage and remains in a sexually immature condition as a neutral individual, sex being an attribute only of the final stage in the development, namely the medusa. The polyp of the Leptolinae has reached the limit of its individual development and is incapable of becoming itself a medusa, but only produces medusa-buds; hence a true alternation of generations is produced. In Trachylinae also the beginnings of a similar metagenesis can be found. Thus in _Cunina octonaria_, the ovum develops into an actinula which buds daughter-actinulae; all of them, both parent and offspring, develop into medusae, so that there is no alternation of generations, but only larval multiplication. In _Cunina parasitica_, however, the ovum develops into an actinula, which buds actinulae as before, but only the daughter-actinulae develop into medusae, while the original, parent-actinula dies off; here, therefore, larval budding has led to a true alternation of generations. In _Gonionemus_ the actinula becomes fixed and polyp-like, and reproduces by budding, so that here also an alternation of generations may occur. In the Leptolinae we must first substitute polyp for actinula, and then a condition is found which can be compared to the case of _Cunina parasitica_ or Gonionemus, if we suppose that neither the parent-actinula (i.e. founder-polyp) nor its offspring by budding (polyps of the colony) have the power of becoming medusae, but only of producing medusae by budding. For further arguments and illustrations the reader must be referred to Brooks's most interesting memoir. The whole theory is one most intimately connected with the question of the relation between polyp and medusa, to be discussed presently. It will be seen elsewhere, however, that whatever view may be held as to the origin of metagenesis in Hydromedusae, in the case of Scyphomedusae (q.v.) no other view is possible than that the alternation of generations is the direct result of larval proliferation.
To complete our survey of life-cycles in the Hydromedusae it is necessary to add a few words about the position of _Hydra_ and its allies. If we accept the view that _Hydra_ is a true sexual polyp, and that its gonads are not gonophores (i.e. medusa-buds) in the extreme of degeneration, then it follows from Brooks's theory that _Hydra_ must be descended from an archaic form in which the medusan type of organization had not yet been evolved. _Hydra_ must, in short, be a living representative of the ancestor of which the actinula-stage is a transient reminiscence in the development of higher forms. It may be pointed out in this connexion that the fixation of _Hydra_ is only temporary, and that the animal is able at all times to detach itself, to move to a new situation, and to fix itself again. There is no difficulty whatever in regarding _Hydra_ as bearing the same relation to the actinula-stage of other Hydromedusae that a Rotifer bears to a trochophore-larva or a fish to a tadpole.
_The Relation of Polyp and Medusa._--Many views have been put forward as to the morphological relationship between the two types of person in the Hydromedusae. For the most part, polyp and medusa have been regarded as modifications of a common type, a view supported by the existence, among Scyphomedusae (q.v.), of sessile polyp-like medusae (_Lucernaria_, &c.). R. Leuckart in 1848 compared medusae in general terms to flattened polyps. G. J. Allman [1] put forward a more detailed view, which was as follows. In some polyps the tentacles are webbed at the base, and it was supposed that a medusa was a polyp of this kind set free, the umbrella being a greatly developed web or membrane extending between the tentacles. A very different theory was enunciated by E. Metchnikoff. In some hydroids the founder-polyp, developed from a planula after fixation, throws out numerous outgrowths from the base to form the hydrorhiza; these outgrowths may be radially arranged so as to form by contact or coalescence a flat plate. Mechnikov considered the plate thus formed at the base of the polyp as equivalent to the umbrella, and the body of the polyp as equivalent to the manubrium, of the medusa; on this view the marginal tentacles almost invariably present in medusae are new formations, and the tentacles of the polyp are represented in the medusa by the oral arms which may occur round the mouth, and which sometimes, e.g. in _Margelidae_, have the appearance and structure of tentacles. Apart from the weighty arguments which the development furnishes against the theories of Allman and Mechnikov, it may be pointed out that neither hypothesis gives a satisfactory explanation of a structure universally present in medusae of whatever class, namely the endoderm-lamella, discovered by the brothers O. and R. Hertwig. It would be necessary to regard this structure as a secondary extension of the endoderm in the tentacle-web, on Allman's theory, or between the outgrowths of the hydrorhiza, on Mechnikov's hypothesis. The development, on the contrary, shows unequivocally that the endoderm-lamella arises as a local coalescence of the endodermal linings of a primitively extensive gastral space.
The question is one intimately connected with the view taken as to the nature and individuality of polyp, medusa and gonophore respectively. On this point the following theories have been put forward.
1. The theory that the medusa is simply an organ, which has become detached and has acquired a certain degree of independence, like the well-known instance of the hectocotyle of the cuttle-fish. On this view, put forward by E. van Beneden and T. H. Huxley, the sporosac is the starting-point of an evolution leading up through the various types of gonophores to the free medusa as the culminating point of a phyletic series. The evidence against this view may be classed under two heads: first, comparative evidence; hydroids very different in their structural characters and widely separate in the systematic classification of these organisms may produce medusae very similar, at least so far as the essential features of medusan organization are concerned; on the other hydroids closely allied, perhaps almost indistinguishable, may produce gonophores in the one case, medusae in the other; for example, _Hydractinia_ (gonophores) and _Podocoryne_ (medusae), _Tubularia_ (gonophores) and _Ectopleura_ (medusae), _Coryne_ (gonophores) and _Syncoryne_ (medusae), and so on. If it is assumed that all these genera bore gonophores ancestrally, then medusa of similar type must have been evolved quite independently in a great number of cases. Secondly, there is the evidence from the development, namely, the presence of the entocodon in the medusa-bud, a structure which, as explained above, can only be accounted for satisfactorily by derivation from a medusan type of organization. Hence it may be concluded that the gonophores are degenerate medusae, and not that the medusae are highly elaborated gonophores, as the organ-theory requires.
2. The theory that the medusa is an independent individual, fully equivalent to the polyp in this respect, is now universally accepted as being supported by all the facts of comparative morphology and development. The question still remains open, however, which of the two types of person may be regarded as the most primitive, the most ancient in the race-history of the Hydromedusae. F. M. Balfour put forward the view that the polyp was the more primitive type, and that the medusa is a special modification of the polyp for reproductive purposes, the result of division of labour in a polyp-colony, whereby special reproductive persons become detached and acquire organs of locomotion for spreading the species. W. K. Brooks, on the other hand, as stated above, regards the medusa as the older type and looks upon both polyp and medusa, in the Hydromedusae, as derived from a free-swimming or floating actinula, the polyp being thus merely a fixed nutritive stage, possessing secondarily acquired powers of multiplication by budding.
The Hertwigs when they discovered the endoderm-lamella showed on morphological grounds that polyp and medusa are independent types, each produced by modification in different directions of a more primitive type represented in development by the actinula-stage. If a polyp, such as _Hydra_, be regarded simply as a sessile actinula, we must certainly consider the polyp to be the older type, and it may be pointed out that in the Anthozoa only polyp-individuals occur. This must not be taken to mean, however, that the medusa is derived from a sessile polyp; it must be regarded as a direct modification of the more ancient free actinula form, without primitively any intervening polyp-stage, such as has been introduced secondarily into the development of the Leptolinae and represents a revival, so to speak, of an ancestral form or larval stage, which has taken on a special role in the economy of the species.
SYSTEMATIC REVIEW OF THE HYDROMEDUSAE
ORDER I. Eleutheroblastea.--Simple polyps which become sexually mature and which also reproduce non-sexually, but without any medusoid stage in the life-cycle.
The sub-order includes the family _Hydridae_, containing the common fresh-water polyps of the genus _Hydra_. Certain other forms of doubtful affinities have also been referred provisionally to this section.
_Hydra._--This genus comprises fresh-water polyps of simple structure. The body bears tentacles, but shows no division into hydrorhiza, hydrocaulus or hydranth; it is temporarily fixed and has no perisarc. The polyp is usually hermaphrodite, developing both ovaries and testes in the same individual. There is no free-swimming planula larva, but the stage corresponding to it is passed over in an enveloping cyst, which is secreted round the embryo by its own ectodermal layer, shortly after the germ-layer formation is complete, i.e. in the parenchymula-stage. The envelope is double, consisting of an external chitinous stratified shell, and an internal thin elastic membrane. Protected by the double envelope, the embryo is set free as a so-called "egg," and in Europe it passes the winter in this condition. In the spring the embryo bursts its shell and is set free as a minute actinula which becomes a _Hydra_.
Many species are known, of which three are common in European waters. It has been shown by C. F. Jickeli (28) that the species are distinguishable by the characters of their nematocysts. They also show characteristic differences in the egg (Brauer [2]). In _Hydra viridis_ the polyp is of a green colour and produces a spherical egg with a smooth shell which is dropped into the mud. _H. grisea_ is greyish in tint and produces a spherical egg with a spiky shell, which also is dropped into the mud. _H. fusca_ (= _H. vulgaris_) is brown in colour, and produces a bun-shaped egg, spiky on the convex surface, and attached to a water-weed or some object by its flattened side. Brauer found a fourth species, similar in appearance to _H. fusca_, but differing from the three other species in being of separate sexes, and in producing a spherical egg with a knobby shell, which is attached like that of _H. fusca_.
The fact already noted that the species of _Hydra_ can be distinguished by the characters of their nematocysts is a point of great interest. In each species, two or three kinds of nematocysts occur, some large, some small, and for specific identification the nematocysts must be studied collectively in each species. It is very remarkable that this method of characterizing and diagnozing species has never been extended to the marine hydroids. It is quite possible that the characters of the nematocysts might afford data as useful to the systematist in this group as do the spicules of sponges, for instance. It would be particularly interesting to ascertain how the nematocysts of a polyp are related to those possessed by the medusa budded from it, and it is possible that in this manner obscure questions of relationship might be cleared up.
_Protohydra_ is a marine genus characterized by the absence of tentacles, by a great similarity to _Hydra_ in histological structure, and by reproduction by transverse fission. It was found originally in an oyster-farm at Ostend. The sexual reproduction is unknown. For further information see C. Chun (HYDROZOA [1]. Pl. I.).
_Polypodium hydriforme_ Ussow is a fresh-water form parasitic on the eggs of the sterlet. A "stolon" of unknown origin produces thirty-two buds, which become as many _Polypodia_; each has twenty-four tentacles and divides by fission repeated twice into four individuals, each with six tentacles. The daughter-individuals grow, form the full number of twenty-four tentacles and divide again. The polyps are free and walk on their tentacles. See Ussow [54].
_Tetraplatia volitans_ Viguier is a remarkable floating marine form. See C. Viguier [56] and Delage and Hérouard (Hydrozoa [2]).
_Haleremita_ Schaudinn. See F. Schaudinn [50] and Delage and Hérouard (HYDROZOA [2]).
In all the above-mentioned genera, with the exception of _Hydra_, the life-cycle is so imperfectly known that their true position cannot be determined in the present state of our knowledge. They may prove eventually to belong to other orders. Hence only the genus _Hydra_ can be considered as truly representing the order Eleutheroblastea. The phylogenetic position of this genus has been discussed above.
ORDER II. Hydroidea seu Leptolinae.--Hydromedusae with alternation of generations (metagenesis) in which a non-sexual polyp-generation (trophosome) produces by budding a sexual medusa-generation (gonosome). The polyp may be solitary, but more usually produces polyps by budding and forms a polyp-colony. The polyp usually has the body distinctly divisible into hydranth, hydrocaulus and hydrorhiza, and is usually clothed in a perisarc. The medusae may be set free or may remain attached to the polyp-colony and degenerate into a gonophore. When fully developed the medusa is characterized by the sense organs being composed entirely of ectoderm, developed independently of the tentacles, and innervated from the sub-umbral nerve-ring.
The two kinds of persons present in the typical Hydroidea make the classification of the group extremely difficult, for reasons explained above. Hence the systematic arrangement that follows must be considered purely provisional. A natural classification of the Hydroidea has yet to be put forward. Many genera and families are separated by purely artificial characters, mere shelf-and-bottle groupings devised, for the convenience of the museum curator and the collector. Thus many subdivisions are diagnosed by setting free medusae in one case, or producing gonophores in another, although it is very obvious, as pointed out above, that a genus producing medusae may be far more closely allied to one producing gonophores than to another producing medusae, or vice versa, and that in some cases the production of medusae or gonophores varies with the season or the sex. Moreover, P. Hallez [22] has recently shown that hydroids hitherto regarded as distinct species are only forms of the same species grown under different conditions.
SUB-ORDER 1. HYDROIDEA GYMNOBLASTEA (ANTHOMEDUSAE).--Trophosome without hydrothecae or gonothecae, with monopodial type of budding. Gonosome with free medusae or gonophores; medusae usually with ocelli, never with otocysts. The gymnoblastic polyp usually has a distinct perisarc investing the hydrorhiza and the hydrocaulus, sometimes also the hydranth as far as the bases of the tentacles (_Bimeria_); but in such cases the perisarc forms a closely-fitting investment or cuticule on the hydranth, never a hydrotheca standing off from it, as in the next sub-order. The polyps may be solitary, or form colonies, which may be of the spreading or encrusting type, or arborescent, and then always of monopodial growth and budding. In some cases, any polyp of the colony may bud medusae; in other cases, only certain polyps, the blastostyles, have this power. When blastostyles are present, however, they are never enclosed in special gonothecae as in the next sub-order. In this sub-order the characters of the hydranth are very variable, probably owing to the fact that it is exposed and not protected by a hydrotheca, as in Calyptoblastea.
Speaking generally, three principal types of hydranth can be distinguished, each with subordinate varieties of form.
1. Club-shaped hydranths with numerous tentacles, generally scattered irregularly, sometimes with a spiral arrangement, or in whorls ("verticillate").
(a) Tentacles filiform; type of _Clava_ (fig. 5), _Cordylophora_, &c.
(b) Tentacles capitate, simple; type of _Coryne_ and _Syncoryne_; _Myriothela_ is an aberrant form with some of the tentacles modified as "claspers" to hold the ova.
(c) Tentacles capitate, branched, wholly or in part; type of _Cladocoryne_.
(d) Tentacles filiform or capitate, tending to be arranged in definite whorls; type of _Stauridium_ (fig. 2), _Cladonema_ and _Pennaria_.
2. Hydranth more shortened, daisy-like in form, with two whorls of tentacles, oral and aboral.
(a) Tentacles filiform, simple, radially arranged or scattered irregularly; type of _Tubularia_ (fig. 4), _Corymorpha_ (fig. 3), _Nemopsis_, _Pelagohydra_, &c.
(b) Tentacles with a bilateral arrangement, branched tentacles in addition to simple filiform ones; type of _Branchiocerianthus_.
3. Hydranth with a single circlet of tentacles.
(a) With filiform tentacles; the commonest type, seen in _Bougainvillea_ (fig. 13), _Eudendrium_, &c.
(b) With capitate tentacles; type of _Clavatella_.
4. Hydranth with tentacles reduced below four; type of _Lar_ (fig. 11), _Monobrachium_, &c.
The _Anthomedusa_ in form is generally deep, bell-shaped. The sense organs are typically ocelli, never otocysts. The gonads are borne on the manubrium, either forming a continuous ring (Codonid type), or four masses or pairs of masses (Oceanid type). The tentacles may be scattered singly round the margin of the umbrella ("monerenematous") or arranged in tufts ("lophonematous"); in form they may be simple or branched (Cladonemid type); in structure they may be hollow ("coelomerinthous"); or solid ("pycnomerinthous"). When sessile gonophores are produced, they may show all stages of degeneration.
_Classification._--Until quite recently the hydroids (Gymnoblastea) and the medusae (Anthomedusae) have been classified separately, since the connexion between them was insufficiently known. Delage and Hérouard (HYDROZOA [2]) were the first to make an heroic attempt to unite the two classifications into one, to which Hickson (HYDROZOA [4]) has made some additions and slight modifications. The classification given here is for the most part that of Delage and Hérouard. It is certain, however, that no such classification can be considered final at present, but must undergo continual revision in the future. With this reservation we may recognize fifteen well-characterized families and others of more doubtful nature. Certain discrepancies must also be noted.
1. _Margelidae_ (= medusa-family _Margelidae_ + hydroid families _Bougainvillidae_, _Dicorynidae_, _Bimeridae_ and _Eudendridae_). Trophosome arborescent, with hydranths of _Bougainvillea_-type; gonosome free medusae or gonophores, the medusae with solid tentacles in tufts (lophonematous). Common genera are the hydroid _Bougainvillea_ (figs. 12, 13), and the medusae _Hippocrene_ (budded from _Bougainvillea_), _Margelis_, _Rathkea_ (fig. 24), and _Margellium_. Other hydroids are _Garveia_, _Bimeria_, _Eudendrium_ and _Heterocordyle_, with gonophores, and _Dicoryne_ with peculiar sporosacs.
2. _Podocorynidae_ (= medusa-families _Thamnostomidae_ and _Cytaeidae_ + hydroid families _Podocorynidae_ and _Hydractiniidae_). Trophosome encrusting with hydranths of _Bougainvillea_-type, polyps differentiated into blastostyles, gastrozoids and dactylozoids; gonosome free medusae or gonophores. The typical genus is the well-known hydroid _Podocoryne_, budding the medusa known as _Dysmorphosa_; _Thamnostylus_, _Cytaeis_, &c., are other medusae with unknown hydroids. _Hydractinia_ (figs. 9, 10) is a familiar hydroid genus, bearing gonophores.
3. _Cladonemidae_.--Trophosome, polyps with two whorls of tentacles, the lower filiform, the upper capitate; gonosome, free medusae, with tentacles solid and branched. The type-genus _Cladonema_ (fig. 20) is a common British form.
4. _Clavatellidae._--Trophosome, polyps with a single whorl of capitate tentacles; gonosome, free medusae, with tentacles branched, solid. _Clavatella_ (fig. 21), with a peculiar ambulatory medusa is a British form.
5. _Pennariidae_.--Trophosome, polyps with an upper circlet of numerous capitate tentacles, and a lower circlet of filiform tentacles. _Pennaria_, with a free medusa known as _Globiceps_, is a common Mediterranean form. _Stauridium_ (fig. 2) is a British hydroid.
6. _Tubulariidae._--Trophosome, polyps with two whorls of tentacles, both filiform. _Tubularia_ (fig. 4), a well-known British hydroid, bears gonophores.
7. _Corymorphidae_ (including the medusa-family _Hybocodonidae_).--Trophosome solitary polyps, with two whorls of tentacles; gonosome, free medusae or gonophores. _Corymorpha_ (fig. 3), a well-known British genus, sets free a medusa known as _Steenstrupia_ (fig. 22). Here belong the deep-sea genera _Monocaulus_ and _Branchiocerianthus_, including the largest hydroid polyps known, both genera producing sessile gonophores.
8. _Dendroclavidae._--Trophosome, polyp with filiform tentacles in three or four whorls. _Dendroclava_, a hydroid, produces the medusa known as _Turritopsis_.
9. _Clavidae_ (including the medusa-family _Tiaridae_ (figs. 27 and 51). Trophosome, polyps with scattered filiform tentacles; gonosome, medusae or gonophores, the medusae with hollow tentacles. _Clava_ (fig. 5), a common British hydroid, produces gonophores; so also does _Cordylophora_, a form inhabiting fresh or brackish water. _Turris_ produces free medusae. _Amphinema_ is a medusan genus of unknown hydroid.
10. _Bythotiaridae._--Trophosome unknown; gonosome, free medusae, with deep, bell-shaped umbrella, with interradial gonads on the base of the stomach, with branched radial canals, and correspondingly numerous hollow tentacles. _Bythotiara_, _Sibogita_.
11. _Corynidae_ (= hydroid families _Corynidae_, _Syncorynidae_ and _Cladocorynidae_ + medusan family _Sarsiidae_).--Trophosome polyps with capitate tentacles, simple or branched, scattered or verticillate; gonosome, free medusae or gonophores. _Coryne_, a common British hydroid, produces gonophores; _Syncoryne_, indistinguishable from it, produces medusae known as _Sarsia_ (fig. 51). _Cladocoryne_ is another hydroid genus; _Codonium_ and _Dipurena_ (fig. 50) are medusan genera.
12. _Myriothelidae._--The genus _Myriothela_ is a solitary polyp with scattered capitate tentacles, producing sporosacs.
13. _Hydrolaridae._--Trophosome (only known in one genus), polyps with two tentacles forming a creeping colony; gonosome, free medusae with four, six or more radial canals, giving off one or more lateral branches which run to the margin of the umbrella, with the stomach produced into four, six or more lobes, upon which the gonads are developed; the mouth with four lips or with a folded margin; the tentacles simple, arranged evenly round the margin of the umbrella. The remarkable hydroid _Lar_ (fig. 11) grows upon the tubes of the worm _Sabella_ and produces a medusa known as _Willia_. Another medusan genus is _Proboscidactyla_.
14. _Monobrachiidae._--The genus _Monobrachium_ is a colony-forming hydroid which grows upon the shells of bivalve molluscs, each polyp having but a single tentacle. It buds medusae, which, however, are as yet only known in an immature condition (C. Mereschkowsky [41]).
15. _Ceratellidae._--Trophosome polyps forming branching colonies of which the stem and main branches are thick and composed of a network of anastomosing coenosarcal tubes covered by a common ectoderm and supported by a thick chitinous perisarc; hydranths similar to those of _Coryne_; gonosome, sessile gonophores. _Ceratella_, an exotic genus from the coast of East Africa, New South Wales and Japan. The genera _Dehitella_ Gray and _Dendrocoryne_ Inaba should perhaps be referred to this family; the last-named is regarded by S. Goto [16] as the type of a distinct family, _Dendrocorynidae_.
Doubtful families, or forms difficult to classify, are: Pteronemidae, Medusae of Cladonemid type, with hydroids for the most part unknown. The British genus _Gemmaria_, however, is budded from a hydroid referable to the family _Corynidae_. _Pteronema_ (fig. 53).
_Nemopsidae_, for the floating polyp _Nemopsis_, very similar to _Tubularia_ in character; the medusa, on the other hand, is very similar to _Hippocrene_ (_Margelidae_). See C. Chun (HYDROZOA [1]).
_Pelagohydridae_, for the floating polyp _Pelagohydra_, Dendy, from New Zealand. The animal is a solitary polyp bearing a great number of medusa-buds. The body, representing the hydranth of an ordinary hydroid, has the aboral portion modified into a float, from which hangs down a proboscis bearing the mouth. The float is covered with long tentacles and bears the medusa-buds. The proboscis bears at its extremity a circlet of smaller oral tentacles. Thus the affinities of the hydranth are clearly, as Dendy points out, with a form such as _Corymorpha_, which also is not fixed but only rooted in the mud. The medusae, on the other hand, have the tentacles in four tufts of (in the buds) five each, and thus resemble the medusae of the family _Margelidae_. See A. Dendy [12].
_Perigonimus._--This common British hydroid belongs by its characters to the family _Bougainvillidae_; it produces, however, a medusa of the genus _Tiara_ (fig. 52), referable to the family _Clavidae_; a fact sufficient to indicate the tentative character of even the most modern classifications of this order.
SUB-ORDER II. HYDROIDEA CALYPTOBLASTEA (LEPTOMEDUSAE).--Trophosome with polyps always differentiated into nutritive and reproductive individuals (blastostyles) enclosed in hydrothecae and gonothecae respectively; with sympodial type of budding. Gonosome with free medusae or gonophores; the medusae typically with otocysts, sometimes with cordyli or ocelli (figs. 54, 55).
The calyptoblastic polyp of the nutritive type is very uniform in character, its tendency to variation being limited, as it were, by the enclosing hydrotheca. The hydranth almost always has a single circlet of tentacles, like the _Bougainvillea_-type, in the preceding sub-order; an exception is the curious genus _Clathrozoon_, in which the hydranth has a single tentacle. The characteristic hydrotheca is formed by the bud at an early stage (fig. 56); when complete it is an open cup, in which the hydranth develops and can be protruded from the opening for the capture of food, or is withdrawn into it for protection. Solitary polyps are unknown in this sub-order; the colony may be creeping or arborescent in form; if the latter, the budding of the polyps, as already stated, is of the sympodial type, and either biserial, forming stems capable of further branching, or uniserial, forming pinnules not capable of further branching. In the biserial type the polyps on the two sides of the stem have primitively an alternating, zigzag arrangement; but, by a process of differential growth, quickened in the 1st, 3rd, 5th, &c., members of the stem, and retarded in the 2nd, 4th, 6th, &c., members, the polyps may assume secondarily positions opposite to one another on the two sides of the stem. Other variations in the mode of growth or budding bring about further differences in the building up of the colony, which are not in all cases properly understood and cannot be described in detail here. The stem may contain a single coenosarcal tube ("monosiphonic") or several united in a common perisarc ("polysiphonic"). An important variation is seen, in the form of the hydrotheca itself, which may come off from the main stem by a stalk, as in _Obelia_, or may be sessile, without a stalk, as in _Sertularia_.
In many Calyptoblastea there occur also reduced defensive polyps or dactylozoids, which in this sub-order have received the special name of _sarcostyles_. Such are the "snake-like zoids" of _Ophiodes_ and other genera, and as such are generally interpreted the "machopolyps" of the _Plumularidea_. These organs are supported by cuplike structures of the perisarc, termed nematophores, regarded as modified hydrothecae supporting the specialized polyp-individuals. They are specially characteristic of the family _Plumularidae_.
The medusa-buds, as already stated, are always produced from blastostyles, reduced non-nutritive polyps without mouth or tentacles. An apparent, but not real, exception is _Halecium halecinum_, in which the blastostyle is produced from the side of a nutritive polyp, and both are enclosed in a common theca without a partition between them (Allman [1] p. 50, fig. 24). The gonotheca is formed in its early stage in the same way as the hydrotheca, but the remains of the hydranth persists as an operculum closing the capsule, to be withdrawn when the medusae or genital products are set free (fig. 56).
The blastostyles, gonophores and gonothecae furnish a series of variations which can best be considered as so many stages of evolution.
Stage 1, seen in _Obelia_. Numerous medusae are budded successively within the gonotheca and set free; they swim off and mature in the open sea (Allman [1], p. 48, figs. 18, 19).
Stage 2, seen in _Gonothyraea_. Medusae, so-called "meconidia," are budded but not liberated; each in turn, when it reaches sexual maturity, is protruded from the gonotheca by elongation of the stalk, and sets free the embryos, after which it withers and is replaced by another (Allman [1], p. 57, fig. 28).
Stage 3, seen in _Sertularia_.--The gonophores are reduced in varying degree, it may be to sporosacs; they are budded successively from the blastostyle, and each in turn, when ripe, protrudes the spadix through the gonotheca (fig. 57, A, B). The spadix forms a gelatinous cyst, the so-called acrocyst (ac), external to the gonotheca (gth), enclosing and protecting the embryos. Then the spadix withers, leaving the embryos in the acrocyst, which may be further protected by a so-called marsupium, a structure formed by tentacle-like processes growing out from the blastostyle to enclose the acrocyst, each such process being covered by perisarc like a glove-finger secreted by it (fig. 57, C). (Allman [1], pp. 50, 51, figs. 21-24; Weismann [58], p. 170, pl. ix., figs. 7, 8.)
Stage 4, seen in _Plumularidae_.--The generative elements are produced in structures termed corbulae, formed by reduction and modification of branches of the colony. Each corbula contains a central row of blastostyles enclosed and protected by lateral rows of branches representing stunted buds (Allman [1], p. 66, fig. 30).
The _Leptomedusa_ in form is generally shallow, more or less saucer-like, with velum less developed than in Anthomedusae (fig. 55). The characteristic sense-organs are ectodermal otocysts, absent, however, in some genera, in which case cordyli may replace them. When otocysts are present, they are at least eight in number, situated adradially, but are often very numerous. The cordyli are scattered on the ring-canal. Ocelli, if present, are borne on the tentacle-bulbs. The tentacles are usually hollow, rarely solid (_Obelia_). In number they are rarely less than four, but in _Dissonema_ there are only two. Primitively there are four perradial tentacles, to which may be added four interradial, or they may become very numerous and are then scattered evenly round the margin, never arranged in tufts or clusters. In addition to tentacles, there may be marginal cirri (_Laodice_) with a solid endodermal axis, spirally coiled, very contractile, and bearing a terminal battery of nematocysts. The gonads are developed typically beneath the radial canals or below the stomach or its pouches, often stretching as long bands on to the base of the manubrium. In _Octorchidae_ (fig. 58) each such band is interrupted, forming one mass at the base of the manubrium and another below the radial canal in each radius, in all eight separate gonad-masses, as the name implies. In some Leptomedusae excretory "marginal tubercles" are developed on the ring-canal.
_Classification._---As in the Gymnoblastea, the difficulty of uniting the hydroid and medusan systems into one scheme of classification is very great in the present state of our knowledge. In a great many Leptomedusae the hydroid stage is as yet unknown, and it is by no means certain even that they possess one. It is quite possible that some of these medusae will be found to be truly hypogenetic, that is to say, with a life-cycle secondarily simplified by suppression of metagenesis. At present, ten recent and one extinct family of Calyptoblastea (Leptomedusae) may be recognized provisionally:
1. _Eucopidae_ (figs. 55, 59).--Trophosome with stalked hydrothecae; gonosome, free medusae with otocysts and four, rarely six or eight, unbranched radial canals. Two of the commonest British hydroids belong to this family, _Obelia_ and _Clytia_. _Obelia_ forms numerous polyserial stems of the characteristic zigzag pattern growing up from a creeping basal stolon, and buds the medusa of the same name. In _Clytia_ the polyps arise singly from the stolon, and the medusa is known as _Phialidium_ (fig. 59).
2. _Aequoridae._--Trophosome only known in one genus (_Polycanna_), and similar to the preceding; gonosome, free medusae with otocysts and with at least eight radial canals, often a hundred or more, simple or branched. _Aequorea_ is a common medusa.
3. _Thaumantidae._--Trophosome only known in one genus (_Thaumantias_), similar to that of the _Eucopidae_; gonosome, free medusae with otocysts inconspicuous or absent, with usually four, sometimes eight, rarely more than eight, radial canals, simple and unbranched, along which the gonads are developed, with numerous tentacles bearing ocelli and with marginal sense-clubs. _Laodice_ and _Thaumantias_ are representative genera.
4. _Berenicidae._--Trophosome unknown; gonosome, free medusae, with four or six radial canals, bearing the gonads, with numerous tentacles, between which occur sense-clubs, without otocysts. _Berenice_, _Staurodiscus_, &c.
5. _Polyorchidae._--Trophosome unknown; gonosome, free medusae of deep form, with radial canals branched in a feathery manner, and bearing gonads on the main canal, but not on the branches, with numerous hollow tentacles bearing ocelli, and without otocysts. _Polyorchis_, _Spirocodon_.
6. _Campanularidae.-_-Trophosome as in _Eucopidae_; gonosome, sessile gonophores. Many common or well-known genera belong here, such as _Halecium_, _Campanularia_, _Gonothyraea_, &c.
7. _Lafoëidae._--Trophosome as in the preceding; gonosome, free medusae or gonophores, the medusae with large open otocysts. The hydroid genus _Lafoëa_ is remarkable for producing gonothecae on the hydrorhiza, each containing a blastostyle which bears a single gonophore; this portion of the colony was formerly regarded as an independent parasitic hydroid, and was named _Coppinia_. Medusan genera are _Mitrocoma_, _Halopsis_, _Tiaropsis_ (fig. 29, &c.).
(So far as the characters of the trophosome are concerned, the seven preceding families are scarcely distinguishable, and they form a section apart, contrasting sharply with the families next to be mentioned, in none of which are free medusae liberated from the colony, so that only the characters of the trophosome need be considered.)
8. _Sertularidae._--Hydrothecae sessile, biserial, alternating or opposite on the stem. _Sertularia_ and _Sertularella_ are two very common genera of this family.
9. _Plumularidae._--Hydrothecae sessile, biserial on the main stem, uniserial on the lateral branches or pinnules, which give the colony its characteristic feathery form; with nematophores. A very abundant and prolific family; well-known British genera are _Plumularia_, _Antennularia_ and _Aglaophenia_.
10. _Hydroceratinidae_.--This family contains the single Australian species _Clathrozoon wilsoni_ Spencer, in which a massive hydrorhiza bears sessile hydrothecae, containing hydranths each with a single tentacle, and numerous nematophores. See W. B. Spencer [53].
11. _Dendrograptidae_, containing fossil (Silurian) genera, such as _Dendrograptus_ and _Thamnograptus_, of doubtful affinities.
ORDER III. Hydrocorallinae.--Metagenetic colony-forming Hydromedusae, in which the polyp-colony forms a massive, calcareous _corallum_ into which the polyps can be retracted; polyp-individuals always of two kinds, gastrozoids and dactylozoids; gonosome either free medusae or sessile gonophores. The trophosome consists of a mass of coenosarcal tubes anastomosing in all planes. The interspaces between the tubes are filled up by a solid mass of lime, consisting chiefly of calcium carbonate, which replaces the chitinous perisarc of ordinary hydroids and forms a stony corallum or _coenosteum_ (fig. 60). The surface of the coenosteum is covered by a layer of common ectoderm, containing large nematocysts, and is perforated by pores of two kinds, gastropores and dactylopores, giving exit to gastrozoids and dactylozoids respectively, which are lodged in vertical pore-canals of wider calibre than the coenosarcal canals of the general network. The coenosteum increases in size by new growth at the surface; and in the deeper, older portions of massive forms the tissues die off after a certain time, only the superficial region retaining its vitality down to a certain depth. The living tissues at the surface are cut off from the underlying dead portions by horizontal partitions termed _tabulae_, which are formed successively as the coenosteum increases in age and size. If the coenosteum of _Millepora_ be broken across, each pore-canal (perhaps better termed a polyp-canal) is seen to be interrupted by a series of transverse partitions, representing successive periods of growth with separation from the underlying dead portions.
Besides the wider vertical pore-canals and the narrower, irregular coenosarcal canals, the coenosteum may contain, in its superficial portion, chambers or _ampullae_, in which the reproductive zoids (medusae or gonophores) are budded from the coenosarc.
The gastropores and dactylopores are arranged in various ways at the surface, a common pattern being the formation of a cyclosystem (fig. 60), in which a central gastrozoid is surrounded by a ring of dactylozoids (fig. 61). In such a system the dactylopores may be confluent with the gastropore, so that the entire cyclosystem presents itself as a single aperture subdivided by radiating partitions, thus having a superficial resemblance to a madreporarian coral with its radiating septa (figs. 62 and 63).
The gastrozoids usually bear short capitate tentacles, four, six or twelve in number; but in _Astylus_ (fig. 63) they have no tentacles. The dactylozoids have no mouth; in _Milleporidae_ they have short capitate tentacles, but lack tentacles in _Stylasteridae_.
The gonosome consists of free medusae in _Milleporidae_, which are budded from the apex of a dactylozoid in _Millepora murrayi_, but in other species from the coenosarcal canals. The medusae are produced by direct budding, without an entocodon in the bud. They are liberated in a mature condition, and probably live but a short time, merely sufficient to spread the species. The manubrium bearing the gonads is mouthless, and the umbrella is without tentacles, sense-organs, velum or radial canals. In the _Stylasteridae_ sessile gonophores are formed, always by budding from the coenosarc. In _Distichopora_ the gonophores have radial canals, but in other genera they are sporosacs with no trace of medusoid structure.
_Classification._---Two families are known:--
1. _Milleporidae._--Coenosteum massive, irregular in form; pores scattered irregularly or in cyclosystems, without styles, with transverse tabulae; free medusae. A single genus, _Millepora_ (figs. 60, 61).
2. _Stylasteridae._--Coenosteum arborescent, sometimes fanlike, with pores only on one face, or on the lateral margins of the branches; gastropores with tabulae only in two genera, but with (except in _Astylus_) a _style_, i.e. a conical, thorn-like projection from the base of the pore, sometimes found also in dactylopores; sessile gonophores. _Sporadopora_ has the pores scattered irregularly. _Distichopora_ has the pores arranged in rows. _Stylaster_ has cyclosystems. In _Allopora_ the cyclostems resemble the calyces of Anthozoan corals. In _Cryptohelia_ the cyclosystem is covered by a cap or operculum. In _Astylus_ (fig. 63) styles are absent.
_Affinities of the Hydrocorallinae._--There can be no doubt that the forms comprised in this order bear a close relationship to the Hydroidea, especially the sub-order Gymnoblastea, with which they should perhaps be classed in a natural classification. A hydrocoralline may be regarded as a form of hydroid colony in which the coenosarc forms a felt-work ramifying in all planes, and in which the chitinous perisarc is replaced by a massive calcareous skeleton. So far as the trophosome is concerned, the step from an encrusting hydroid such as _Hydractinia_ to the hydrocoralline _Millepora_ is not great.
Hickson considers that the families _Milleporidae_ and _Stylasteridae_ should stand quite apart from one another and should not be united in one order. The nearest approach to the _Stylasteridae_ is perhaps to be found in _Ceratella_, with its arborescent trophosome formed of anastomosing coenosarcal tubes supported by a thick perisarc and covered by a common ectoderm. _Ceratella_ stands in much the same relation to the _Stylasteridae_ that _Hydractinia_ does to the _Milleporidae_, in both cases the chitinous perisarc being replaced by the solid coenosteum to which the hydrocorallines owe the second half of their name.
ORDER IV. Graptolitoidea (Rhabdophora, Allman).--This order has been constituted for a peculiar group of palaeozoic fossils, which have been interpreted as the remains of the skeletons of Hydrozoa of an extinct type.
A typical graptolite consists of an axis bearing a series of tooth-like projections, like a saw. Each such projection is regarded as representing a cup or hydrotheca, similar to those borne by a calyptoblastic hydroid, such as _Sertularia_. The supposed hydrothecae may be present on one side of the axis only (monoprionid) or on both sides (diprionid); the first case may be conjectured to be the result of uniserial (helicoid) budding, the second to be produced by biserial (scorpioid) budding. In one division (_Retiolitidae_) the axis is reticulate. In addition to the stems bearing cups, there are found vesicles associated with them, which have been interpreted as gonothecae or as floats, that is to say, air-bladders, acting as hydrostatic organs for a floating polyp-colony.
Since no graptolites are known living, or, indeed, since palaeozoic times, the interpretation of their structure and affinities must of necessity be extremely conjectural, and it is by no means certain that they are Hydrozoa at all. It can only be said that their organization, so far as the state of their preservation permits it to be ascertained, offers closer analogies with the Hydrozoa, especially the Calyptoblastea, than with any other existing group of the animal kingdom.
See the treatise of Delage and Hérouard (HYDROZOA, [4]), and the article GRAPTOLITES.
ORDER V. Trachylinea.--Hydromedusae without alternation of generations, i.e. without a hydroid phase; the medusa develops directly from the actinula larva, which may, however, multiply by budding. Medusae with sense-organs represented by otocysts derived from modified tentacles (tentaculocysts), containing otoliths of endodermal origin, and innervated from the ex-umbral nerve-ring.
This order, containing the typical oceanic medusae, is divided into two sub-orders.
SUB-ORDER 1. TRACHOMEDUSAE.--Tentacles given off from the margin of the umbrella, which is entire, i.e. not lobed or indented; tentaculocysts usually enclosed in vesicles; gonads on the radial canals. The medusae of this order are characterized by the tough, rigid consistence of the umbrella, due partly to the dense nature of the mesogloea, partly to the presence of a marginal rim of chondral tissue, consisting of thickened ectoderm containing great numbers of nematocysts, and forming, as it were, a cushion-tyre supporting the edge of the umbrella. Prolongations from the rim of chondral tissue may form clasps or _peronia_ supporting the tentacles. The tentacles are primarily four in number, perradial, alternating with four interradial tentaculocysts, but both tentacles and sense-organs may be multiplied and the primary perradii may be six instead of four (fig. 26). The tentacles are always solid, containing an axis of endoderm-cells resembling notochordal tissue or plant-parenchyma, and are but moderately flexible. The sense-organs are tentaculocysts which are usually enclosed in vesicles and may be sunk far below the surface. The gonads are on the radial canals or on the stomach (_Ptychogastridae_), and each gonad may be divided into two by a longitudinal sub-umbral muscle-tract. The radial canals are four, six, eight or more, and in some genera blindly-ending centripetal canals are present (fig. 26). The stomach may be drawn out into the manubrium, forming a proboscis ("Magenstiel") of considerable length.
The development of the Trachomedusae, so far as it is known, shows an actinula-stage which is either free (larval) or passed over in the egg (foetal) as in _Geryonia_; in no case does there appear to be a free planula-stage. The actinula, when free, may multiply by larval budding, but in all cases both the original actinula and all its descendants become converted into medusae, so that there is no alternation of generations. In _Gonionemus_ the actinula becomes attached and polyp-like and reproduces by budding.
The Trachomedusae are divided into the following families:
1. _Petasidae_ (_Petachnidae_).--Four radial canals, four gonads; stomach not prolonged into the manubrium, which is relatively short; tentaculocysts free. _Petasus_ and other genera make up this family, founded by Haeckel, but no other naturalist has ever seen them, and it is probable that they are simply immature forms of other genera.
2. _Olindiadae_, with four radial canals and four gonads; manubrium short; ring-canals giving off blind centripetal canals; tentaculocysts enclosed. _Olindias mülleri_ (fig. 64) is a common Mediterranean species. Other genera are _Aglauropsis_, _Gossea_ and _Gonionemus_; the last named bears adhesive suckers on the tentacles. Some doubt attaches to the position of this family. It has been asserted that the tentaculocysts are entirely ectodermal and that either the family should be placed amongst the Leptomedusae, or should form, together with certain Leptomedusae, an entirely distinct order. In _Gonionemus_, however, the concrement-cells are endodermal.
3. _Trachynemidae._--Eight radial canals, eight gonads, stomach not prolonged into manubrium; tentaculocysts enclosed. _Rhopalonema_, _Trachynema_, &c.
4. _Ptychogastridae_ (_Pectyllidae_).--As in the preceding, but with suckers on the tentacles. _Ptychogastria_ Allman (=_Pectyllis_), a deep-sea form.
5. _Aglauridae._--Eight radial canals, two, four or eight gonads; tentacles numerous; tentaculocysts free; stomach prolonged into manubrium. _Aglaura_, _Aglantha_ (fig. 65), &c., with eight gonads; _Stauraglaura_ with four; _Persa_ with two. _Amphogona_, hermaphrodite, with male and female gonads on alternating radial canals.
6. _Geryonidae._--Four or six radial canals; gonads band-like; stomach prolonged into a manubrium of great length; tentaculocysts enclosed. _Liriope_, &c., with four radial canals; _Geryonia_, _Carmarina_ (fig. 26), &c., with six.
7. _Halicreidae._--Eight very broad radial canals; ex-umbrella often provided with lateral outgrowths; tentacles differing in size, but in a single row. _Halicreas_.
SUB-ORDER 2. NARCOMEDUSAE.--Margin of the umbrella-lobed, tentacles arising from the ex-umbrella at some distance from the margin; tentaculocysts exposed, not enclosed in vesicles; gonads on the sub-umbral floor of the stomach or of the gastric pouches.
The Narcomedusae exhibit peculiarities of form and structure which distinguish them at once from all other Hydromedusae. The umbrella is shallow and has the margin supported by a rim of thickened ectoderm, as in the Trachomedusae, but not so strongly developed. The tentacles are not inserted on the margin of the umbrella, but arise high up on the ex-umbral surface, and the umbrella is prolonged into lobes corresponding to the interspaces between the tentacles. The condition of things can be imagined by supposing that in a medusa primitively of normal build, with tentacles at the margin, the umbrella has grown down past the insertion of the tentacles. As a result of this extension of the umbrellar margin, all structures belonging to this region, namely, the ring-canal, the nerve-rings, and the rim of thickened ectoderm, do not run an even course, but are thrown into festoons, caught up under the insertion of each tentacle in such a way that the ring-canal and its accompaniments form in each notch of the umbrellar margin an inverted V, the apex of which corresponds to the insertion of the tentacle; in some cases the limbs of the V may run for some distance parallel to one another, and may be fused into one, giving a figure better compared to an inverted Y. Thus the ectodermal rim runs round the edge of each lobe of the umbrella and then passes upwards towards the base of the tentacle from the re-entering angle between two adjacent lobes, to form with its fellow of the next lobe a tentacle-clasp or _peronium_, i.e. a streak of thickened ectoderm supporting the tentacle. Similarly the ring-canal runs round the edge of the lobe as the so-called festoon-canal, and then runs upwards under the peronium to the base of the tentacle as one of a pair of peronial canals, the limbs of the V-like figure already mentioned. The nerve-rings have a similar course. The tentaculocysts are implanted round the margins of the lobes of the umbrella and may be supported by prolongations of the ectodermal rim termed _otoporpae_ (_Gehörspangen_). The radial canals are represented by wide gastric pouches, and may be absent, so that the tentacles arise directly from the stomach (_Solmaridae_). The tentacles are always solid, as in Trachomedusae.
The development of the Narcomedusae is in the main similar to that of the Trachomedusae, but shows some remarkable features. In _Aeginopsis_ a planula is formed by multipolar immigration. The two ends of the planula become greatly lengthened and give rise to the two primary tentacles of the actinula, of which the mouth arises from one side of the planula. Hence the principal axis of the future medusa corresponds, not to the longitudinal axis of the planula, but to a transverse axis. This is in some degree parallel to the cases described above, in which a planula gives rise to the hydrorhiza, and buds a polyp laterally.
In _Cunina_ and allied genera the actinula, formed in the manner described, has a hypostome of great length, quite disproportionate to the size of the body, and is further endowed with the power of producing buds from a stolon arising from the aboral side of the body. In these species the actinula is parasitic upon another medusa; for instance, _Cunoctantha octonaria_ upon _Turritopsis_, _C. proboscidea_ upon _Liriope_ or _Geryonia_. The parasite effects a lodgment in the host either by invading it as a free-swimming planula, or, apparently, in other cases, as a spore-embryo which is captured and swallowed as food by the host. The parasitic actinula is found attached to the proboscis of the medusa; it thrusts its greatly elongated hypostome into the mouth of the medusa and nourishes itself upon the food in the digestive cavity of its host. At the same time it produces buds from an aboral stolon. The buds become medusae by the direct method of budding described above. In some cases the buds do not become detached at once, but the stolon continues to grow and to produce more buds, forming a "bud-spike" (_Knospenähre_), which consists of the axial stolon bearing medusa-buds in all stages of development. In such cases the original parent-actinula does not itself become a medusa, but remains arrested in development and ultimately dies off, so that a true alternation of generations is brought about. It is in these parasitic forms that we meet with the method of reproduction by sporogony described above.
In other Narcomedusae, e.g. _Cunoctantha fowleri_ Browne, buds are formed from the sub-umbrella on the under side of the stomach pouches, where later the gonads are developed.
_Classification._--Three families of Narcomedusae are recognized (see O. Maas [40]):
1. _Cunanthidae._--With broad gastric pouches which are simple, i.e. undivided, and "pernemal," i.e. correspond in position with the tentacles. _Cunina_ (fig. 66) with more than eight tentacles; _Cunoctantha_ with eight tentacles, four perradial, four interradial.
2. _Aeginidae._--Radii a multiple of four, with radial gastric pouches bifurcated or subdivided; the tentacles are implanted in the notch between the two subdivisions of each (primary) gastric pouch, hence the (secondary) gastric pouches appear to be "internemal" in position, i.e. to alternate in position with the tentacles. _Aegina_, with four tentacles and eight pouches; _Aeginura_ (fig. 25), with eight tentacles and sixteen pouches; _Solmundella_ (fig. 67), with two tentacles and eight pouches; _Aeginopsis_ (fig. 23), with two or four tentacles and sixteen pouches.
3. _Solmaridae._--No gastric pouches; the numerous tentacles arise direct from the stomach, into which also the peronial canals open, so that the ring-canal is cut up into separate festoons. _Solmaris_, _Pegantha_, _Polyxenia_, &c. To this family should be referred, probably, the genus _Hydroctena_, described by C. Dawydov [11a] and regarded by him as intermediate between Hydromedusae and Ctenophora. See O. Maas [35].
_Appendix to the Trachylinae._
Of doubtful position, but commonly referred to the Trachylinae, are the two genera of fresh-water medusae, _Limnocodium_ and _Limnocnida_.
_Limnocodium sowerbyi_ was first discovered in the _Victoria regia_ tank in the Botanic Gardens, Regent's Park, London. Since then it has been discovered in other botanic gardens in various parts of Europe, its two most recent appearances being at Lyons (1901) and Munich (1905), occurring always in tanks in which the _Victoria regia_ is cultivated, a fact which indicates that tropical South America is its original habitat. In the same tanks a small hydroid, very similar to _Microhydra_, has been found, which bears medusa-buds and is probably the stock from which the medusa is budded. It is a remarkable fact that all specimens of _Limnocodium_ hitherto seen have been males; it may be inferred from this either that only one polyp-stock has been introduced into Europe, from which all the medusae seen hitherto have been budded, or perhaps that the female medusa is a sessile gonophore, as in _Pennaria_. The male gonads are carried on the radial canals.
_Limnocnida tanganyicae_ was discovered first in Lake Tanganyika, but has since been discovered also in Lake Victoria and in the river Niger. It differs from _Limnocodium_ in having practically no manubrium but a wide mouth two-thirds the diameter of the umbrella across. It buds medusae from the margin of the mouth in May and June, and in August and September the gonads are formed in the place where the buds arose. The hydroid phase, if any, is not known.
Both these medusae have sense-organs of a peculiar type, which are said to contain an endodermal axis like the sense-organs of Trachylinae, but the fact has recently been called in question for _Limnocodium_ by S. Goto, who considers the genus to be allied to _Olindias_. Allman, on the other hand, referred _Limnocodium_ to the Leptomedusae.
In this connexion must be mentioned, finally, the medusae budded from the fresh-water polyp _Microhydra_. The polyp-stages of _Limnocodium_ and _Microhydra_ are extremely similar in character. In both cases the hydranth is extremely reduced and has no tentacles, and the polyp forms a colony by budding from the base. In _Limnocodium_ the body secretes a gelatinous mucus to which adhere particles of mud, &c., forming a protective covering. In _Microhydra_ no such protecting case is formed. In view of the great resemblance between _Microhydra_ and the polyp of _Limnocodium_, it might be expected that the medusae to which they give origin would also be similar. As yet, however, the medusa of _Microhydra_ has only been seen in an immature condition, but it shows some well-marked differences from _Limnocodium_, especially in the structure of the tentacles, which furnish useful characters for distinguishing species amongst medusae. The possession of a polyp-stage by _Limnocodium_ and _Microhydra_ furnishes an argument against placing them in the Trachylinae. Their sense-organs require renewed investigations. (Browne [10] and [10a].)
ORDER VI. Siphonophora.--Pelagic floating Hydrozoa with great differentiation of parts, each performing a special function; generally regarded as colonies showing differentiation of individuals in correspondence with a physiological division of labour.
A typical Siphonophore is a stock or _cormus_ consisting of a number of _appendages_ placed in organic connexion with one another by means of a _coenosarc_. The coenosarc does not differ in structure from that already described in colonial Hydrozoa. It consists of a hollow tube, or tubes, of which the wall is made up of the two body-layers, ectoderm and endoderm, and the cavity is a continuation of the digestive cavities of the nutritive and other appendages, i.e. of the coelenteron. The coenosarc may consist of a single elongated tube or stolon, forming the stem or axis of the cormus on which, usually, the appendages are arranged in groups termed cormidia; or it may take the form of a compact mass of ramifying, anastomosing tubes, in which case the cormus as a whole has a compact form and _cormidia_ are not distinguishable. In the Disconectae the coenosarc forms a spongy mass, the "_centradenia_," which is partly hepatic in function, forming the so-called liver, and partly excretory.
The appendages show various types of form and structure corresponding to different functions. The cormus is always differentiated into two parts; an upper portion termed the _nectosome_, in which the appendages are locomotor or hydrostatic in function, that is to say, serve for swimming or floating; and a lower portion termed the _siphosome_, bearing appendages which are nutritive, reproductive or simply protective in function.
Divergent views have been held by different authors both as regards the nature of the cormus as a whole, and as regards the homologies of the different types of appendages borne by it.
The general theories of Siphonophoran morphology are discussed below, but in enumerating the various types of appendages it is convenient to discuss their morphological interpretation at the same time.
In the nectosome one or more of the following types of appendage occur:--
1. Swimming-bells, termed _nectocalyces_ or _nectophores_ (fig. 68, k), absent in _Chondrophorida_ and _Cystophorida_; they are contractile and resemble, both in appearance, structure and function, the umbrella of a medusa, with radial canals, ring-canal and velum; but they are without manubrium, tentacles or sense-organs, and are always bilaterally symmetrical, a peculiarity of form related with the fact that they are attached on one side to the stem. A given cormus may bear one or several nectocalyces, and by their contractions they propel the colony slowly along, like so many medusae harnessed together. In cases where the cormus has no pneumatophore the topmost swimming bell may contain an oil-reservoir or _oleocyst_.
2. The pneumatophore or air-bladder (fig. 68, n), for passive locomotion, forming a float which keeps the cormus at or near the surface of the water. The pneumatophore arises from the ectoderm as a pit or invagination, part of which forms a gas-secreting gland, while the rest gives rise to an air-sack lined by a chitinous cuticle. The orifice of invagination forms a pore which may be closed up or may form a protruding duct or funnel. As in the analogous swim-bladder of fishes, the gas in the pneumatophore can be secreted or absorbed, whereby the specific gravity of the body can be diminished or increased, so as to cause it to float nearer the surface or at a deeper level. Never more than one pneumatophore is found in a cormus, and when present it is always situated at the highest point above the swimming bells, if these are present also. In _Velella_ the pneumatophore becomes of complex structure and sends air-tubes, lined by a chitin and resembling tracheae, down into the compact coenosarc, thus evidently serving a respiratory as well as a hydrostatic function.
Divergent views have been held as to the morphological significance of the pneumatophore. E. Haeckel regarded the whole structure as a glandular ectodermal pit formed on the ex-umbral surface of a medusa-person. C. Chun and, more recently, R. Woltereck [59], on the other hand, have shown that the ectodermal pit which gives rise to the pneumatophore represents an entocodon. Hence the cavity of the air-sack is equivalent to a sub-umbral cavity in which no manubrium is formed, and the pore or orifice of invagination would represent the margin of the umbrella. In the wall of the sack is a double layer of endoderm, the space between which is a continuation of the coelenteron. By coalescence of the endoderm-layers, the coelenteron may be reduced to vessels, usually eight in number, opening into a ring-sinus surrounding the pore. Thus the disposition of the endoderm-cavities is roughly comparable to the gastrovascular system of a medusa.
The difference between the theories of Haeckel and Chun is connected with a further divergence in the interpretation of the stem or axis of the cormus. Haeckel regards it as the equivalent of the manubrium, and as it is implanted on the blind end of the pneumatophore, such a view leads necessarily to the air-sack and gland being a development on the ex-umbral surface of the medusa-person. Chun and Woltereck, on the other hand, regard the stem as a _stolo prolifer_ arising from the aboral pole, that is to say, from the ex-umbrella, similar to that which grows out from the ex-umbral surface of the embryo of the Narcomedusae and produces buds, a view which is certainly supported by the embryological evidence to be adduced shortly.
In the siphosome the following types of appendages occur:--
1. _Siphons_ or nutritive appendages, from which the order takes its name; never absent and usually present in great numbers (fig. 68, e). Each is a tube dilated at or towards the base and containing a mouth at its extremity, leading into a stomach placed in the dilatation already mentioned. The siphons have been compared to the manubrium of a medusa-individual, or to polyps, and hence are sometimes termed gastrozoids.
2. _Palpons_ (fig. 68, g), present in some genera, especially in Physonectae; similar to the siphons but without a mouth, and purely tactile in function, hence sometimes termed dactylozoids. If a distal pore or aperture is present, it is excretory in function; such varieties have been termed "cystons" by Haeckel.
3. _Tentacles_ ("_Fangfäden_"), always present, and implanted one at the base of each siphon (fig. 68, f). The tentacles of siphonophores may reach a great length and have a complex structure. They may bear accessory filaments or _tentilla_ (f'), covered thickly with batteries of nematocysts, to which these organisms owe their great powers of offence and defence.
4. _Palpacles_ ("_Tastfäden_"), occurring together with palpons, one implanted at the base of each palpon (fig. 68, h). Each palpacle is a tactile filament, very extensile, without accessory filaments or nematocysts.
5. _Bracts_ ("_hydrophyllia_"), occur in _Calycophorida_ and some _Physophorida_ as scale-like appendages protecting other parts (fig. 68, l). The mesogloea is greatly developed in them and they are often of very tough consistency. By Haeckel they are considered homologous with the umbrella of a medusa.
6. _Gonostyles_, appendages which produce by budding medusae or gonophores, like the blastostyles of a hydroid colony. In their most primitive form they are seen in _Velella_ as "gonosiphons," which possess mouths like the ordinary sterile siphons and bud free medusae. In other forms they have no mouths. They may be branched, so-called "gonodendra," and amongst them may occur special forms of palpons, "gonopalpons." The gonostyles have been compared to the blastostyles of a hydroid colony, or to the manubrium of a medusa which produces free or sessile medusa-buds.
7. _Gonophores_, produced either on the gonostyles already mentioned or budded, as in hydrocorallines, from the coenosarc, i.e. the stem (fig. 68, i.). They show every transition between free medusae and sporosacs, as already described, for hydroid colonies. Thus in _Velella_ free medusae are produced, which have been described as an independent genus of medusae, _Chrysomitra_. In other types the medusae may be set free in a mature condition as the so-called "genital swimming bells," comparable to the _Globiceps of Pennaria_. The most usual condition, however, is that in which sessile medusoid gonophores or sporosacs are produced.
The various types of appendages described in the foregoing may be arranged in groups termed _cormidia_. In forms with a compact coenosarc such as _Velella_, _Physalia_, &c., the separate cormidia cannot be sharply distinguished, and such a condition is described technically as one with "scattered" cormidia. In forms in which, on the other hand, the coenosarc forms an elongated, tubular axis or stem, the appendages are arranged as regularly recurrent cormidia along it, and the cormidia are then said to be "ordinate." In such cases the oldest cormidia, that is to say, those furthest from the nectosome, may become detached (like the segments or proglottides of a tape-worm) and swim off, each such detached cormidium then becoming a small free cormus which, in many cases, has been given an independent generic name. A cormidium may contain a single nutritive siphon ("monogastric") or several siphons ("polygastric"):
The following are some of the forms of cormidia that occur:--
1. The _eudoxome_ (Calycophorida), consisting of a bract, siphon, tentacle and gonophore; when free it is known as _Eudoxia_.
2. The _ersaeome_ (Calycophorida), made up of the same appendages as the preceding type but with the addition of a nectocalyx; when free termed _Ersaea_.
3. The _rhodalome_ of some _Rhodalidae_, consisting of siphon, tentacle and one or more gonophores.
4. The _athorome_ of _Physophora_, &c., consisting of siphon, tentacle, one or more palpons with palpacles, and one or more gonophores.
5. The _crystallome_ of _Anthemodes_, &c., similar to the athorome but with the addition of a group of bracts.
_Embryology of the Siphonophora._--The fertilized ovum gives rise to a parenchymula, with solid endoderm, which is set free as a free-swimming planula larva, in the manner already described (see HYDROZOA). The planula has its two extremities dissimilar (Bipolaria-larva). The subsequent development is slightly different according as the future cormus is headed by a pneumatophore (Physophorida, Cystophorida) or by a nectocalyx (Calycophorida).
(i.) Physophorida, for example _Halistemma_ (C. Chun, HYDROZOA [1]). The planula becomes elongated and broader towards one pole, at which a pit or invagination of the ectoderm arises. Next the pit closes up to form a vesicle with a pore, and so gives rise to the pneumatophore. From the broader portion of the planula an outgrowth arises which becomes the first tentacle of the cormus. The endoderm of the planula now acquires a cavity, and at the narrower pole a mouth is formed, giving rise to the primary siphon. Thus from the original planula three appendages are, as it were, budded off, while the planula itself mostly gives rise to coenosarc, just as in some hydroids the planula is converted chiefly into hydrorhiza.
(ii.) Calycophorida, for example, _Muggiaea_. The planula develops, on the whole, in a similar manner, but the ectodermal invagination arises, not at the pole of the planula, but on the side of its broader portion, and gives rise, not to a pneumatophore, but to a nectocalyx, the primary swimming bell or _protocodon_ ("_Fallschirm_") which is later thrown off and replaced by secondary swimming bells, _metacodons_, budded from the coenosarc.
From a comparison of the two embryological types there can be no doubt on two points; first, that the pneumatophore and the protocodon are strictly homologous, and, therefore if the nectocalyx is comparable to the umbrella of a medusa, as seems obvious, the pneumatophore must be so too; secondly, that the coenosarcal axis arises from the ex-umbrella of the medusa and cannot be compared to a manubrium, but is strictly comparable to the "bud-spike" of a Narcomedusan.
_Theories of Siphonophore Morphology._--The many theories that have been put forward as to the interpretation of the cormus and the various parts are set forth and discussed in the treatise of Y. Delage and E. Hérouard (HYDROZOA [4]) and more recently by R. Woltereck [59], and only a brief analysis can be given here.
In the first place the cormus has been regarded as a single individual and its appendages as _organs_. This is the so-called "polyorgan" theory, especially connected with the name of Huxley; but it must be borne in mind that Huxley regarded all the forms produced, in any animal, between one egg-generation and the next, as constituting in the lump one single individual. Huxley, therefore, considered a hydroid colony, for example, as a single individual, and each separate polyp or medusa budded from it as having the value of an organ and not of an individual. Hence Huxley's view is not so different from those held by other authors as it seems to be at first sight.
In more recent years Woltereck [59] has supported Huxley's view of individuality, at the same time drawing a fine distinction between "individual" and "person." The individual is the product of sexual reproduction; a person is an individual of lower rank, which may be produced asexually. A Siphonophore is regarded as a single individual composed of numerous zoids, budded from the primary zoid (siphon) produced from the planula. Any given zoid is a person-zoid if equivalent to the primary zoid, an organ-zoid if equivalent only to a part of it. Woltereck considers the siphonophores most nearly allied to the Narcomedusae, producing like the buds from an aboral stolon, the first bud being represented by the pneumatophore or protocodon, in different cases.
Contrasting, in the second place, with the polyorgan theory are the various "polyperson" theories which interpret the Siphonophore cormus as a colony composed of more or fewer individuals in organic union with one another. On this interpretation there is still room for considerable divergence of opinion as regards detail. To begin with, it is not necessary on the polyperson theory to regard each appendage as a distinct individual; it is still possible to compare appendages with parts of an individual which have become separated from one another by a process of "dislocation of organs." Thus a bract may be regarded, with Haeckel, as a modified umbrella of a medusa, a siphon as its manubrium, and a tentacle as representing a medusan tentacle shifted in attachment from the margin to the sub-umbrella; or a siphon may be compared with a polyp, of which the single tentacle has become shifted so as to be attached to the coenosarc and so on. Some authors prefer, on the other hand, to regard every appendage as a separate individual, or at least as a portion of an individual, of which other portions have been lost or obliterated.
A further divergence of opinion arises from differences in the interpretation of the persons composing the colony. It is possible to regard the cormus (1) as a colony of medusa-persons, (2) as a colony of polyp-persons, (3) as composed partly of one, partly of the other. It is sufficient here to mention briefly the views put forward on this point by C. Chun and R. Haeckel.
Chun (HYDROZOA [1]) maintains the older views of Leuckart and Claus, according to which the cormus is to be compared to a floating hydroid colony. It may be regarded as derived from floating polyps similar to _Nemopsis_ or _Pelagohydra_, which by budding produce a colony of polyps and also form medusa-buds. The polyp-individuals form the nutritive siphosome or trophosome. The medusa-buds are either fertile or sterile. If fertile they become free medusae or sessile gonophores. If sterile they remain attached and locomotor in function, forming the nectosome, the pneumatophore and swimming-bells.
Haeckel, on the other hand, is in accordance with Balfour in regarding a Siphonophore as a medusome, that is to say, as a colony composed of medusoid persons or organs entirely. Haeckel considers that the Siphonophores have two distinct ancestral lines of evolution:
1. In the _Disconanthae_, i.e. in such forms as _Velella_, _Porpita_, &c., the ancestor was an eight-rayed medusa (_Disconula_) which acquired a pneumatophore as an ectodermal pit on the ex-umbrella, and in which the organs (manubrium, tentacles, &c.) became secondarily multiplied, just as they do in _Gastroblasta_ as the result of incomplete fission. The nearest living allies of the ancestral _Disconula_ are to be sought in the _Pectyllidae_.
2. In the _Siphonanthae_, i.e. in all other Siphonophores, the ancestral form was a _Siphonula_, a bilaterally symmetrical Anthomedusa with a single long tentacle (cf. _Corymorpha_), which became displaced from the margin to the sub-umbrella. The _Siphonula_ produced buds on the manubrium, as many Anthomedusae are known to do, and these by reduction or dislocation of parts gave rise to the various appendages of the colony. Thus the umbrella of the _Siphonula_ became the protocodon, and its manubrium, the axis or stolon, which, by a process of dislocation of organs, escaped, as it were, from the sub-umbrella through a cleft and became secondarily attached to the ex-umbrella. It must be pointed out that, however probable Haeckel's theory may be in other respects, there is not the slightest evidence for any such cleft in the umbrella having been present at any time, and that the embryological evidence, as already pointed out, is all against any homology between the stem and a manubrium, since the primary siphon does not become the stem, which arises from the ex-umbral side of the protocodon and is strictly comparable to a stolon.
_Classification._--The Siphonophora may be divided, following Delage and Hérouard, into four sub-orders:
I. CHONDROPHORIDA (_Disconectae_ Haeckel, _Tracheophysae_ Chun). With an apical chambered pneumatophore, from which tracheal tubes may take origin (fig. 70); no nectocalyces or bracts; appendages all on the lower side of the pneumatophore arising from a compact coenosarc, and consisting of a central principal siphon, surrounded by gonosiphons, and these again by tentacles.
Three families: (1) _Discalidae_, for _Discalia_ and allied genera, deep-sea forms not well known; (2) _Porpitidae_ for the familiar genus _Porpita_ (fig. 69) and its allies; and (3) _Velellidae_, represented by the well-known genus _Velella_ (figs. 70, 71), common in the Mediterranean and other seas.
II. CALYCOPHORIDA (_Calyconectae_, Haeckel). Without pneumatophore, with one, two, rarely more nectocalyces.
Three families: (1) _Monophyidae_, with a single nectocalyx; examples _Muggiaea_, sometimes found in British seas, _Sphaeronectes_, &c.; (2) _Diphyidae_, with two nectocalyces; examples _Diphyes_ (fig. 72), _Praya_, _Abyla_, &c.; and (3) _Polyphyidae_, with numerous nectocalyces; example _Hippopodius_, _Stephanophyes_ and other genera.
III. PHYSOPHORIDA (_Physonectae_ + _Auronectae_, Haeckel). With an apical pneumatophore, not divided into chambers, followed by a series of nectocalyces or bracts.
A great number of families and genera are referred to this group, amongst which may be mentioned specially--(1) _Agalmidae_, containing the genera _Stephanomia_, _Agalma_, _Anthemodes_, _Halistemma_, &c.; (2) _Apolemidae_, with the genus _Apolemia_ and its allies; (3) _Forskaliidae_, with _Forskalia_ and allied forms; (4) _Physophoridae_, for _Physophora_ (fig. 73) and other genera, (5) _Anthophysidae_, for _Anthophysa_, _Athorybia_, &c.; and lastly the two families (6) _Rhodalidae_ and (7) _Stephalidae_ (fig. 74), constituting the group Auronectae of Haeckel. The Auronectae are peculiar deep-sea forms, little known except from Haeckel's descriptions, in which the large pneumatophore has a peculiar duct, termed the aurophore, placed on its lower side in the midst of a circle of swimming-bells.
IV. CYSTOPHORIDA (_Cystonectae_, Haeckel). With a very large pneumatophore not divided into chambers, but without nectocalyces or bracts. Two sections can be distinguished, the Rhizophysina, with long tubular coenosarc-bearing ordinate cormidia, and Physalina, with compact coenosarc-bearing scattered cormidia.
A type of the Rhizophysina is the genus _Rhizophysa_. The Physalina comprise the families _Physalidae_ and _Epibulidae_, of which the types are _Physalia_ (figs. 74, 75) and _Epibulia_, respectively. _Physalia_, known commonly as the Portuguese man-of-war, is remarkable for its great size, its brilliant colours, and its terrible stinging powers.
BIBLIOGRAPHY.--In addition to the works cited below, see the general works cited in the article HYDROZOA, in some of which very full bibliographies will be found.
1. G. J. Allman, "A Monograph of the Gymnoblastic or Tubularian Hydroids," Ray Society (1871-1872); 2. A. Brauer, "Über die Entwickelung von Hydra," _Zeitschr. f. wiss. Zool._ lii. (1891), pp. 169-216, pls. ix.-xii.; 3. "Über die Entstehung der Geschlechtsprodukte und die Entwickelung von Tubularia mesembryanthemum Allm.," _t.c._ pp. 551-579, pls. xxxiii.-xxxv.; 4. W. K. Brooks, "The Life-History of the Hydromedusae: a discussion of the Origin of the Medusae, and of the significance of Metagenesis," _Mem. Boston Soc. Nat. Hist._ iii. (1886), pp. 259-430, pis. xxxvii.-xliv.; 5. "The Sensory Clubs of Cordyli of _Laodice_," _Journ. Morphology_, x. (1895), pp. 287-304, pl. xvii.; 6. E. T. Browne, "On British Hydroids and Medusae," _Proc. Zool. Soc._ (1896), pp. 459-500, pls. xvi., xvii., (1897), pp. 816-835, pls. xlviii. xlix. 12 text-figs.; 7. "Biscayan Medusae," _Trans. Linn. Soc._ x. (1906), pp. 163-187, pl. xiii.; 8. "Medusae" in Herdman, _Rep. Pearl Oyster Fisheries, Gulf of Manaar_, iv. (1905), pp. 131-166, 4 pls.; 9. "Hydromedusae with a Revision of the _Williadae_ and _Petasidae_," _Fauna and Geogr. Maldive and Laccadive Archipelagos_, ii. (1904), pp. 722-749, pls. liv.-lvii.; 10. "On the Freshwater Medusa liberated by _Microhydra ryderi_, Potts, and a Comparison with _Limnocodium_," _Quart. Journ. Micr. Sci._ I (1906), pp. 635, 645, pl. xxxvii.; 10a. "On the Freshwater Medusa _Limnocnida tanganicae_" _Budgett Memorial Volume_ (Cambridge, 1908, pp. 471-482, pl. xxviii.; 11. C. Claus, "Über die Struktur der Muskelzellen und über den Körperbau von Mnestra parasites Krohn," _Verhandl. zool. bot. Ges. Wien_, xxv. (1876), pp. 9-12, pl. i.; 11a. C. Dawydov, "Hydroctena salenskii," _Mém. Acad. Imp. St. Pétersbourg_ (viii.) xiv. No. 9 (1903), 17 pp., 1 pl.; 12. A. Dendy, "On a Free-swimming Hydroid, _Pelagohydra mirabilis_," n. gen. et sp., _Quart. Journ. Micr. Sci._ xlvi. (1903), pp. 1-24, pls. i. ii.; 13. H. Driesch, "Tektonische Studien an Hydroidpolypen," (1) _Jen. Zeitschr._, xxiv. (1890), pp. 189-226, 12 figs.; (2) _t.c._ pp. 657-688, 6 figs.; (3) _ibid._ xxv. (1891), pp. 467-479, 3 figs.; 14. G. Duplessis, "On _Campanularia volubilis_," _Soc. Vaud. Bull._ 13 (Lausanne, 1874-1875); 15. J. W. Fewkes, "On _Mnestra_," _Amer. Natural._, xviii. (1884), pp. 197-198, 3 figs.; 16. S. Goto, "Dendrocoryne Inaba, Vertreterin einer neuen Familie der Hydromedusen," _Annot. Zool. Tokyo_, i. (1897), pp. 93-104, pl. vi., figs. 106-113; 17. "The Craspe dote Medusa _Olindias_ and some of its Natural Allies," _Mark Anniversary Volume_ (New York, 1903), pp. 1-22, 3 pls.; 18. H. Grenacher, "Über die Nesselkapseln von Hydra," _Zool. Anz._ xviii. (1895), pp. 310-321, 7 figs.; 19. R. T. Günther, "On the Structure and Affinities of _Mnestra parasites_ Krohn; with a revision of the Classification of the _Cladonemidae_," _Mitt. Stat. Neapel_, xvi. (1903), pp. 35-62, pls. ii. iii.; 20. E. Haeckel, "Das System der Medusen," _Denkschr. med.-nat.-wiss. Ges._ (Jena, 1879-1881); 21. "Deep Sea Medusae," in _Reports of the Challenger Expedition_, Zool. iv. pt. 2 (London, 1882); 22. P. Hallez, "Bougainvillia fruticosa Allm. est le faciès d'eau agitée du Bougainvillia ramosa Van Ben." _C.-R. Acad. Sci. Paris_, cxl. (1905), pp. 457-459; 23. O. & R. Hertwig, _Der Organismus der Medusen_ (Jena, 1878), 70 pp., 3 pls.; 24. _Das Nervensystem und die Sinnesorgane der Medusen_ (Leipzig, 1878), 186 pp., 10 pls.; 25. S. J. Hickson, "The Medusae of _Millepora_," _Proc. Roy. Soc._ lxvi. (1899), pp. 6-10, 10 figs.; 26. T. Hincks, _A History of British Hydroid Zoophytes_ (2 vols., London, 1868); 27. N. Iwanzov, "Über den Bau, die Wirkungsweise und die Entwickelung der Nesselkapseln von Coelenteraten," _Bull. Soc. Imp. Natural, Moscou_ (1896), pp. 323-355, 4 pls.; 28. C. F. Jickeli, "Der Bau der Hydroidpolypen," (1) _Morph. Jahrbuch_, viii. (1883), pp. 373-416, pls. xvi.-xviii.; (2) t.c., pp. 580-680, pls. xxv.-xxviii.; 29. Albert Lang, "Über die Knospung bei Hydra und einigen Hydropolypen," _Zeitschr. f. wiss. Zool._ liv. (1892), pp. 365-384, pl. xvii.; 30. Arnold Lang, "Gastroblasta Raffaelei. Eine durch eine Art unvollständiger Theilung entstehende Medusen-Kolonie," _Jena Zeitschr._ xix. (1886), pp. 735-762, pls. xx., xxi.; 31. A. Linko, "Observations sur les méduses de la mer Blanche," _Trav. Soc. Imp. Nat. St Pétersbourg_, xxix. (1899); 32. "Über den Bau der Augen bei den Hydromedusen," _Zapiski Imp. Akad. Nauk (Mém. Acad. Imp. Sci.) St Pétersbourg_ (8) x. 3 (1900), 23 pp., 2 pls.; 33. O. Maas, "Die craspedoten Medusen," in _Ergebn. Plankton Expedition_, ii. (Kiel and Leipzig, 1893), 107 pp., 8 pls., 3 figs.; 34. "Die Medusen," _Mem. Mus. Comp. Zool. Harvard_, xxiii. (1897), i.; 35. "On _Hydroctena_," _Zool. Centralbl._ xi. (1904), pp. 240-243; 36. "Revision des méduses appartenant aux familles des _Cunanthidae_ et des _Aeginidae_, et groupement nouveau des genres," _Bull. Mus. Monaco_, v. (1904), 8 pp.; 37. "Revision der Cannotiden Haeckels," _SB. K. Bayer. Akad._ xxxiv. (1904), pp. 421-445; 38. "Meduses," _Result. Camp. Sci. Monaco_, xxviii. (1904), 71 pp., 6 pls.; 39. "Die craspedoten Medusen der Siboga-Expedition," _Uitkomst. Siboga-Exped._ x. (1905), 84 pp., 14 pls.; 40. "Die arktischen Medusen (ausschliesslich der Polypomedusen)," _Fauna arctica_, iv. (1906), pp. 479-526; 41. C. Mereschkowsky, "On a new Genus of Hydroids (_Monobrachium_) from the White Sea, with a short description of other new Hydroids," _Ann. Mag. Nat. Hist._ (4) xx. (1877), pp. 220-229, pls. v. vi.; 42. E. Metchinkoft, "Studien über die Entwickelung der Medusen und Siphonophoren," _Zeitschr. f. wiss. Zool._ xxiv. (1874), pp. 15-83, pls. i.-xii.; 43. "Vergleichend-embryologische Studien" (_Geryoniden, Cunina_), _ibid._ xxxvi. (1882), pp. 433-458, pl. xxviii.; 44. _Embryologische Studien an Medusen_ (Vienna, 1886), 150 pp., 12 pls., 10 figs.; 45. "Medusologische Mittheilungen," _Arb. zool. Inst. Wien_, vi. (1886), pp. 237-266, pls. xxii. xxiii.; 46. L. Murbach, "Beiträge zur Kenntnis der Anatomie und Entwickelung der Nesselorgane der Hydroiden," _Arch. f. Naturgesch._ lx. i. (1894), pp. 217-254, pl. xii.; 47. "Preliminary Note on the Life-History of _Gonionemus_," _Journ. Morph._ xi. (1895), pp. 493-496; 48. L. Murbach and C. Shearer, "On Medusae from the Coast of British Columbia and Alaska," _Proc. Zool. Soc._ (1903), ii. pp. 164-191, pls. xvii.-xxii.; 49. H. F. Perkins, "The Development of _Gonionema murbachii_," _Proc. Acad. Nat. Sci. Philadelphia_ (1902), pp. 750-790, pls. xxxi-xxxiv.; 50. F. Schaudinn, "Über Haleremita cumulans, n. g. n. sp., einen marinen Hydroidpolypen," _SB. Ges. natforsch. Freunde Berlin_ (1894), pp. 226-234, 8 figs.; 51. F. E. Schulze, "On the Structure and Arrangement of the Soft Parts in _Euplectella aspergillum_" (_Amphibrachium_), _Tr. R. Soc. Edinburgh_, xxix. (1880), pp. 661-673, pl. xvii.; 52. O. Seeliger, "Über das Verhalten der Keimblätter bei der Knospung der Cölenteraten," _Zeitschr. f. wiss. Zool._ lviii. (1894), pp. 152-188, pls. vii.-ix.; 53. W. B. Spencer, "A new Family of Hydroidea (_Clathrozoon_), together with a description of the Structure of a new Species of _Plumularia_," _Trans. Roy. Soc. Victoria_ (1890), pp. 121-140, 7 pls.; 54. M. Ussow, "A new Form of Fresh-water Coelenterate" (_Polypodium_), _Ann. Mag. Nat. Hist._ (5) xviii. (1886), pp. 110-124, pl. iv.; 55. E. Vanhöffen, "Versuch einer natürlichen Gruppierung der Anthomedusen," _Zool. Anzeiger_, xiv. (1891), pp. 439-446; 56. C. Viguier, "Études sur les animaux inférieurs de la baie d'Alger" (_Tetraplatia_), _Arch. Zool. Exp. Gen._ viii. (1890), pp. 101-142, pls. vii.-ix.; 57. J. Wagner, "Recherches sur l'organisation de Monobrachium parasiticum Méréjk," _Arch. biol._ x. (1890), pp. 273-309, pls. viii. ix.; 58. A. Weismann, _Die Entstehung der Sexualzellen bei den Hydromedusen_ (Jena, 1883); 59. R. Woltereck, "Beiträge zur Ontogenie und Ableitung des Siphonophorenstocks," _Zeitschr. f. wiss. Zool._ lxxxii. (1905), pp. 611-637, 21 text-figs.; 60. J. Wulfert, "Die Embryonalentwickelung von Gonothyraea loveni Allm.," _Zeitschr. f. wiss. Zool._ lxxi. (1902), pp. 296-326, pls. xvi.-xviii. (E. A. M.)
FOOTNOTES:
[1] In some cases hydroids have been reared in aquaria from ova of medusae, but these hydroids have not yet been found in the sea (Browne [10 a]).
[2] The numbers in square brackets [] refer to the bibliography at the end of this article; but when the number is preceded by the word Hydrozoa, it refers to the bibliography at the end of the article HYDROZOA.
HYDROMETER (Gr. [Greek: hydôr], water, and [Greek: metron], a measure), an instrument for determining the density of bodies, generally of fluids, but in some cases of solids. When a body floats in a fluid under the action of gravity, the weight of the body is equal to that of the fluid which it displaces (see HYDROMECHANICS). It is upon this principle that the hydrometer is constructed, and it obviously admits of two modes of application in the case of fluids: either we may compare the weights of floating bodies which are capable of displacing the same volume of different fluids, or we may compare the volumes of the different fluids which are displaced by the same weight. In the latter case, the densities of the fluids will be inversely proportional to the volumes thus displaced.
The hydrometer is said by Synesius Cyreneus in his fifth letter to have been invented by Hypatia at Alexandria,[1] but appears to have been neglected until it was reinvented by Robert Boyle, whose "New Essay Instrument," as described in the _Phil. Trans._ for June 1675, differs in no essential particular from Nicholson's hydrometer. This instrument was devised for the purpose of detecting counterfeit coin, especially guineas and half-guineas. In the first section of the paper (_Phil. Trans._ No. 115, p. 329) the author refers to a glass instrument exhibited by himself many years before, and "consisting of a bubble furnished with a long and slender stem, which was to be put into several liquors, to compare and estimate their specific gravities." This seems to be the first reference to the hydrometer in modern times.
In fig. 1 C represents the instrument used for guineas, the circular plates A representing plates of lead, which are used as ballast when lighter coins than guineas are examined. B represents "a small glass instrument for estimating the specific gravities of liquors," an account of which was promised by Boyle in the following number of the _Phil. Trans._, but did not appear.
The instrument represented at B (fig. 1), which is copied from Robert Boyle's sketch in the _Phil. Trans._ for 1675, is generally known as the common hydrometer. It is usually made of glass, the lower bulb being loaded with mercury or small shot which serves as ballast, causing the instrument to float with the stem vertical. The quantity of mercury or shot inserted depends upon the density of the liquids for which the hydrometer is to be employed, it being essential that the whole of the bulb should be immersed in the heaviest liquid for which the instrument is used, while the length and diameter of the stem must be such that the hydrometer will float in the lightest liquid for which it is required. The stem is usually divided into a number of equal parts, the divisions of the scale being varied in different instruments, according to the purposes for which they are employed.
Let V denote the volume of the instrument immersed (i.e. of liquid displaced) when the surface of the liquid in which the hydrometer floats coincides with the lowest division of the scale, A the area of the transverse section of the stem, l the length of a scale division, n the number of divisions on the stem, and W the weight of the instrument. Suppose the successive divisions of the scale to be numbered 0, 1, 2 ... n starting with the lowest, and let w0, W1, w2 ... w_n be the weights of unit volume of the liquids in which the hydrometer sinks to the divisions 0, 1, 2 ... n respectively. Then, by the principle of Archimedes,
W = Vw0; or w0 = W/V. Also
W = (V + lA)w1; or w1 = W/(V + lA),
w_p = W/(V + plA), and
w_n= W/(V + nlA),
or the densities of the several liquids vary inversely as the respective volumes of the instrument immersed in them; and, since the divisions of the scale correspond to equal increments of volume immersed, it follows that the densities of the several liquids in which the instrument sinks to the successive divisions form a harmonic series.
If V = NlA then N expresses the ratio of the volume of the instrument up to the zero of the scale to that of one of the scale-divisions. If we suppose the lower part of the instrument replaced by a uniform bar of the same sectional area as the stem and of volume V, the indications of the instrument will be in no respect altered, and the bottom of the bar will be at a distance of N scale-divisions below the zero of the scale.
In this case we have w_p = W/(N + p)lA; or the density of the liquid varies inversely as N + p, that is, as the whole number of scale-divisions between the bottom of the tube and the plane of flotation.
If we wish the successive divisions of the scale to correspond to equal increments in the density of the corresponding liquids, then the volumes of the instrument, measured up to the successive divisions of the scale, must form a series in harmonical progression, the lengths of the divisions increasing as we go up the stem.
The greatest density of the liquid for which the instrument described above can be employed is W/V, while the least density is W/(V + nlA), or W/(V + v), where v represents the volume of the stem between the extreme divisions of the scale. Now, by increasing v, leaving W and V unchanged, we may increase the range of the instrument indefinitely. But it is clear that if we increase A, the sectional area of the stem, we shall diminish l, the length of a scale-division corresponding to a given variation of density, and thereby proportionately diminish the sensibility of the instrument, while diminishing the section A will increase l and proportionately increase the sensibility, but will diminish the range over which the instrument can be employed, unless we increase the length of the stem in the inverse ratio of the sectional area. Hence, to obtain great sensibility along with a considerable range, we require very long slender stems, and to these two objections apply in addition to the question of portability; for, in the first place, an instrument with a very long stem requires a very deep vessel of liquid for its complete immersion, and, in the second place, when most of the stem is above the plane of flotation, the stability of the instrument when floating will be diminished or destroyed. The various devices which have been adopted to overcome this difficulty will be described in the account given of the several hydrometers which have been hitherto generally employed.
The plan commonly adopted to obviate the necessity of inconveniently long stems is to construct a number of hydrometers as nearly alike as may be, but to load them differently, so that the scale-divisions at the bottom of the stem of one hydrometer just overlap those at the top of the stem of the preceding. By this means a set of six hydrometers, each having a stem rather more than 5 in. long, will be equivalent to a single hydrometer with a stem of 30 in. But, instead of employing a number of instruments differing only in the weights with which they are loaded, we may employ the same instrument, and alter its weight either by adding mercury or shot to the interior (if it can be opened) or by attaching weights to the exterior. These two operations are not quite equivalent, since a weight added to the interior does not affect the volume of liquid displaced when the instrument is immersed up to a given division of the scale, while the addition of weights to the exterior increases the displacement. This difficulty may be met, as in Keene's hydrometer, by having all the weights of precisely the same volume but of different masses, and never using the instrument except with one of these weights attached.
The first hydrometer intended for the determination of the densities of liquids, and furnished with a set of weights to be attached when necessary, was that constructed by Mr Clarke (instrument-maker) and described by J. T. Desaguliers in the _Philosophical Transactions_ for March and April 1730, No. 413, p. 278. The following is Desaguliers's account of the instrument (fig. 2):--
"After having made several fruitless trials with ivory, because it imbibes spirituous liquors, and thereby alters its gravity, he (Mr Clarke) at last made a copper hydrometer, represented in fig. 2, having a brass wire of about 1 in. thick going through, and soldered into the copper ball Bb. The upper part of this wire is filed flat on one side, for the stem of the hydrometer, with a mark at m, to which it sinks exactly in proof spirits. There are two other marks, A and B, at top and bottom of the stem, to show whether the liquor be (1/10)th above proof (as when it sinks to A), or (1/10)th under proof (as when it emerges to B), when a brass weight such as C has been screwed on to the bottom at c. There are a great many such weights, of different sizes, and marked to be screwed on instead of C, for liquors that differ more than (1/10)th from proof, so as to serve for the specific gravities in all such proportions as relate to the mixture of spirituous liquors, in all the variety made use of in trade. There are also other balls for showing the specific gravities quite to common water, which make the instrument perfect in its kind."
Clarke's hydrometer, as afterwards constructed for the purposes of the excise, was provided with thirty-two weights to adapt it to spirits of different specific gravities, and eleven smaller weights, or "weather weights" as they were called, which were attached to the instrument in order to correct for variations of temperature. The weights were adjusted for successive intervals of 5° F., but for degrees intermediate between these no additional correction was applied. The correction for temperature thus afforded was not sufficiently accurate for excise purposes, and William Speer in his essay on the hydrometer (Tilloch's _Phil. Mag._, 1802, vol. xiv.) mentions cases in which this imperfect compensation led to the extra duty payable upon spirits which were more than 10% over proof being demanded on spirits which were purposely diluted to below 10% over proof in order to avoid the charge. Clarke's hydrometer, however, remained the standard instrument for excise purposes from 1787 until it was displaced by that of Sikes.
Desaguliers himself constructed a hydrometer of the ordinary type for comparing the specific gravities of different kinds of water (Desaguliers's _Experimental Philosophy_, ii. 234). In order to give great sensibility to the instrument, the large glass ball was made nearly 3 in. in diameter, while the stem consisted of a wire 10 in. in length and only (1/40)in. in diameter. The instrument weighed 4000 grains, and the addition of a grain caused it to sink through an inch. By altering the quantity of shot in the small balls the instrument could be adapted for liquids other than water.
To an instrument constructed for the same purpose, but on a still larger scale than that of Desaguliers, A. Deparcieux added a small dish on the top of the stem for the reception of the weights necessary to sink the instrument to a convenient depth. The effect of weights placed in such a dish or pan is of course the same as if they were placed within the bulb of the instrument, since they do not alter the volume of that part which is immersed.
The first important improvement in the hydrometer after its reinvention by Boyle was introduced by G. D. Fahrenheit, who adopted the second mode of construction above referred to, arranging his instrument so as always to displace the same volume of liquid, its weight being varied accordingly. Instead of a scale, only a single mark is placed upon the stem, which is very slender, and bears at the top a small scale pan into which weights are placed until the instrument sinks to the mark upon its stem. The volume of the displaced liquid being then always the same, its density will be proportional to the whole weight supported, that is, to the weight of the instrument together with the weights required to be placed in the scale pan.
Nicholson's hydrometer (fig. 3) combines the characteristics of Fahrenheit's hydrometer and of Boyle's essay instrument.[2] The following is the description given of it by W. Nicholson in the _Manchester Memoirs_, ii. 374:--
"AA represents a small scale. It may be taken off at D. Diameter 1½ in., weight 44 grains.
"B a stem of hardened steel wire. Diameter 1/100 in.
"E a hollow copper globe. Diameter 2(8/10) in. Weight with stem 369 grains.
"FF a stirrup of wire screwed to the globe at C.
"G a small scale, serving likewise as a counterpoise. Diameter 1½ in. Weight with stirrup 1634 grains.
"The other dimensions may be had from the drawing, which is one-sixth of the linear magnitude of the instrument itself.
"In the construction it is assumed that the upper scale shall constantly carry 1000 grains when the lower scale is empty, and the instrument sunk in distilled water at the temperature of 60° Fahr. to the middle of the wire or stem. The length of the stem is arbitrary, as is likewise the distance of the lower scale from the surface of the globe. But, the length of the stem being settled, the lower scale may be made lighter, and, consequently, the globe less, the greater its distance is taken from the surface of the globe; and the contrary."
In comparing the densities of different liquids, it is clear that this instrument is precisely equivalent to that of Fahrenheit, and must be employed in the same manner, weights being placed in the top scale only until the hydrometer sinks to the mark on the wire, when the specific gravity of the liquid will be proportional to the weight of the instrument together with the weights in the scale.
In the subsequent portion of the paper above referred to, Nicholson explains how the instrument may be employed as a thermometer, since, fluids generally expanding more than the solids of which the instrument is constructed, the instrument will sink as the temperature rises.
To determine the density of solids heavier than water with this instrument, let the solid be placed in the upper scale pan, and let the weight now required to cause the instrument to sink in distilled water at standard temperature to the mark B be denoted by w, while W denotes the weight required when the solid is not present. Then W - w is the weight of the solid. Now let the solid be placed in the lower pan, care being taken that no bubbles of air remain attached to it, and let w1 be the weight now required in the scale pan. This weight will exceed w in consequence of the water displaced by the solid, and the weight of the water thus displaced will be W1 - w, which is therefore the weight of a volume of water equal to that of the solid. Hence, since the weight of the solid itself is W - w, its density must be (W - w)/(w1 - w).
The above example illustrates how Nicholson's or Fahrenheit's hydrometer may be employed as a weighing machine for small weights.
In all hydrometers in which a part only of the instrument is immersed, there is a liability to error in consequence of the surface tension, or capillary action, as it is frequently called, along the line of contact of the instrument and the surface of the liquid (see CAPILLARY ACTION). This error diminishes as the diameter of the stem is reduced, but is sensible in the case of the thinnest stem which can be employed, and is the chief source of error in the employment of Nicholson's hydrometer, which otherwise would be an instrument of extreme delicacy and precision. The following is Nicholson's statement on this point:--
"One of the greatest difficulties which attends hydrostatical experiments arises from the attraction or repulsion that obtains at the surface of the water. After trying many experiments to obviate the irregularities arising from this cause, I find reason to prefer the simple one of carefully wiping the whole instrument, and especially the stem, with a clean cloth. The weights in the dish must not be esteemed accurate while there is either a cumulus or a cavity in the water round the stem."
It is possible by applying a little oil to the upper part of the bulb of a common or of a Sikes's hydrometer, and carefully placing it in pure water, to cause it to float with the upper part of the bulb and the whole of the stem emerging as indicated in fig. 4, when it ought properly to sink almost to the top of the stem, the surface tension of the water around the circumference of the circle of contact, AA', providing the additional support required.
The universal hydrometer of G. Atkins, described in the _Phil. Mag._ for 1808, xxxi. 254, is merely Nicholson's hydrometer with the screw at C projecting through the collar into which it is screwed, and terminating in a sharp point above the cup G. To this point soft bodies lighter than water (which would float if placed in the cup) could be attached, and thus completely immersed. Atkins's instrument was constructed so as to weigh 700 grains, and when immersed to the mark on the stem in distilled water at 60° F. it carried 300 grains in the upper dish. The hydrometer therefore displaced 1000 grains of distilled water at 60° F. and hence the specific gravity of any other liquid was at once indicated by adding 700 to the number of grains in the pan required to make the instrument sink to the mark on the stem. The small divisions on the scale corresponded to differences of (1/10)th of a grain in the weight of the instrument.
The "Gravimeter," constructed by Citizen Guyton and described in _Nicholson's Journal_, 4to, i. 110, differs from Nicholson's instrument in being constructed of glass, and having a cylindrical bulb about 21 centimetres in length and 22 millimetres in diameter. Its weight is so adjusted that an additional weight of 5 grammes must be placed in the upper pan to cause the instrument to sink to the mark on the stem in distilled water at the standard temperature. The instrument is provided with an additional piece, or "plongeur," the weight of which exceeds 5 grammes by the weight of water which it displaces; that is to say, it is so constructed as to weigh 5 grammes in water, and consists of a glass envelope filled with mercury. It is clear that the effect of this "plongeur," when placed in the lower pan, is exactly the same as that of the 5 gramme weight in the upper pan. Without the extra 5 grammes the instrument weighs about 20 grammes, and therefore floats in a liquid of specific gravity .8. Thus deprived of its additional weight it may be used for spirits. To use the instrument for liquids of much greater density than water additional weights must be placed in the upper pan, and the "plongeur" is then placed in the lower pan for the purpose of giving to the instrument the requisite stability.
Charles's balance areometer is similar to Nicholson's hydrometer, except that the lower basin admits of inversion, thus enabling the instrument to be employed for solids lighter than water, the inverted basin serving the same purpose as the pointed screw in Atkins's modification of the instrument.
Adie's sliding hydrometer is of the ordinary form, but can be adjusted for liquids of widely differing specific gravities by drawing out a sliding tube, thus changing the volume of the hydrometer while its weight remains constant.
The hydrometer of A. Baumé, which has been extensively used in France, consists of a common hydrometer graduated in the following manner. Certain fixed points were first determined upon the stem of the instrument. The first of these was found by immersing the hydrometer in pure water, and marking the stem at the level of the surface. This formed the zero of the scale. Fifteen standard solutions of pure common salt in water were then prepared, containing respectively 1, 2, 3, ... 15% (by weight) of dry salt. The hydrometer was plunged in these solutions in order, and the stem having been marked at the several surfaces, the degrees so obtained were numbered 1, 2, 3, ... 15. These degrees were, when necessary, repeated along the stem by the employment of a pair of compasses till 80 degrees were marked off. The instrument thus adapted to the determination of densities exceeding that of water was called the hydrometer for salts.
The hydrometer intended for densities less than that of water, or the hydrometer for spirits, is constructed on a similar principle. The instrument is so arranged that it floats in pure water with most of the stem above the surface. A solution containing 10% of pure salt is used to indicate the zero of the scale, and the point at which the instrument floats when immersed in distilled water at 10° R. (54½° F.) is numbered 10. Equal divisions are then marked off upwards along the stem as far as the 50th degree.
The densities corresponding to the several degrees of Baumé's hydrometer are given by Nicholson (_Journal of Philosophy_, i. 89) as follows:--
_Baumé's Hydrometer for Spirits. Temperature 10° R._
+--------+--------+--------+--------+--------+--------+ |Degrees.|Density.|Degrees.|Density.|Degrees.|Density.| +--------+--------+--------+--------+--------+--------+ | 10 | 1.000 | 21 | .922 | 31 | .861 | | 11 | .990 | 22 | .915 | 32 | .856 | | 12 | .985 | 23 | .909 | 33 | .852 | | 13 | .977 | 24 | .903 | 34 | .847 | | 14 | .970 | 25 | .897 | 35 | .842 | | 15 | .963 | 26 | .892 | 36 | .837 | | 16 | .955 | 27 | .886 | 37 | .832 | | 17 | .949 | 28 | .880 | 38 | .827 | | 18 | .943 | 29 | .874 | 39 | .822 | | 19 | .935 | 30 | .867 | 40 | .817 | | 20 | .928 | | | | | +--------+--------+--------+--------+--------+--------+
_Baume's Hydrometer for Salts._
+--------+--------+--------+--------+--------+--------+ |Degrees.|Density.|Degrees.|Density.|Degrees.|Density.| +--------+--------+--------+--------+--------+--------+ | 0 | 1.000 | 27 | 1.230 | 51 | 1.547 | | 3 | 1.020 | 30 | 1.261 | 54 | 1.594 | | 6 | 1.040 | 33 | 1.295 | 57 | 1.659 | | 9 | 1.064 | 36 | 1.333 | 60 | 1.717 | | 12 | 1.089 | 39 | 1.373 | 63 | 1.779 | | 15 | 1.114 | 42 | 1.414 | 66 | 1.848 | | 18 | 1.140 | 45 | 1.455 | 69 | 1.920 | | 21 | 1.170 | 48 | 1.500 | 72 | 2.000 | | 24 | 1.200 | | | | | +--------+--------+--------+--------+--------+--------+
Carrier's hydrometer was very similar to that of Baumé, Cartier having been employed by the latter to construct his instruments for the French revenue. The point at which the instrument floated in distilled water was marked 10° by Cartier, and 30° on Carrier's scale corresponded to 32° on Baumé's.
Perhaps the main object for which hydrometers have been constructed is the determination of the value of spirituous liquors, chiefly for revenue purposes. To this end an immense variety of hydrometers have been devised, differing mainly in the character of their scales.
In Speer's hydrometer the stem has the form of an octagonal prism, and upon each of the eight faces a scale is engraved, indicating the percentage strength of the spirit corresponding to the several divisions of the scale, the eight scales being adapted respectively to the temperature 35°, 40°, 45°, 50°, 55°, 60°, 65° and 70° F. Four small pins, which can be inserted into the counterpoise of the instrument, serve to adapt the instrument to the temperatures intermediate between those for which the scales are constructed. William Speer was supervisor and chief assayer of spirits in the port of Dublin. For a more complete account of this instrument see Tilloch's _Phil. Mag._, xiv. 151.
The hydrometer constructed by Jones, of Holborn, consists of a spheroidal bulb with a rectangular stem (fig. 5). Between the bulb and counterpoise is placed a thermometer, which serves to indicate the temperature of the liquid, and the instrument is provided with three weights which can be attached to the top of the stem. On the four sides of the stem AD are engraved four scales corresponding respectively to the unloaded instrument, and to the instrument loaded with the respective weights. The instrument when unloaded serves for the range from 74 to 47 over proof; when loaded with the first weight it indicates from 46 to 13 over proof, with the second weight from 13 over proof to 29 under proof, and with the third from 29 under proof to pure water, the graduation corresponding to which is marked W at the bottom of the fourth scale. One side of the stem AD is shown in fig. 5, the other three in fig. 6. The thermometer is also provided with four scales corresponding to the scales above mentioned. Each scale has its zero in the middle corresponding to 60° F. If the mercury in the thermometer stand above this zero the spirit must be reckoned weaker than the hydrometer indicates by the number on the thermometer scale level with the top of the mercury, while if the thermometer indicate a temperature lower than the zero of the scale (60° F.) the spirit must be reckoned stronger by the scale reading. At the side of each of the four scales on the stem of the hydrometer is engraved a set of small numbers indicating the contraction in volume which would be experienced if the requisite amount of water (or spirit) were added to bring the sample tested to the proof strength.
The hydrometer constructed by Dicas of Liverpool is provided with a sliding scale which can be adjusted for different temperatures, and which also indicates the contraction in volume incident on bringing the spirit to proof strength. It is provided with thirty-six different weights which, with the ten divisions on the stem, form a scale from 0 to 370. The employment of so many weights renders the instrument ill-adapted for practical work where speed is an object.
This instrument was adopted by the United States in 1790, but was subsequently discarded by the Internal Revenue Service for another type. In this latter form the observations have to be made at the standard temperature of 60° F., at which the graduation 100 corresponds to proof spirit and 200 to absolute alcohol. The need of adjustable weights is avoided by employing a set of five instruments, graduated respectively 0°-100°, 80°-120°, 100°-140°, 130°-170°, 160°-200°. The reading gives the volume of proof spirit equivalent to the volume of liquor; thus the readings 80° and 120° mean that 100 volumes of the test liquors contain the same amount of absolute alcohol as 80 and 120 volumes of proof spirit respectively. Proof spirit is defined in the United States as a mixture of alcohol and water which contains equal volumes of alcohol and water at 60° F., the alcohol having a specific gravity of 0.7939 at 60° as compared with water at its maximum density. The specific gravity of proof spirit is 0.93353 at 60°; and 100 volumes of the mixture is made from 50 volumes of absolute alcohol and 53.71 volumes of water.
Quin's universal hydrometer is described in the _Transactions of the Society of Arts_, viii. 98. It is provided with a sliding rule to adapt it to different temperatures, and has four scales, one of which is graduated for spirits and the other three serve to show the strengths of worts. The peculiarity of the instrument consists in the pyramidal form given to the stem, which renders the scale-divisions more nearly equal in length than they would be on a prismatic stem.
Atkins's hydrometer, as originally constructed, is described in _Nicholson's Journal_, 8vo, ii. 276. It is made of brass, and is provided with a spheroidal bulb the axis of which is 2 in. in length, the conjugate diameter being 1½ in. The whole length of the instrument is 8 in., the stem square of about 1/8-in. side, and the weight about 400 grains. It is provided with four weights, marked 1, 2, 3, 4, and weighing respectively 20, 40, 61 and 84 grains, which can be attached to the shank of the instrument at C (fig. 7) and retained there by the fixed weight B. The scale engraved upon one face of the stem contains fifty-five divisions, the top and bottom being marked 0 or zero and the alternate intermediate divisions (of which there are twenty-six) being marked with the letters of the alphabet in order. The four weights are so adjusted that, if the instrument floats with the stem emerging as far as the lower division 0 with one of the weights attached, then replacing the weight by the next heavier causes the instrument to sink through the whole length of the scale to the upper division 0, and the first weight produces the same effect when applied to the naked instrument. The stem is thus virtually extended to five times its length, and the number of divisions increased practically to 272. When no weight is attached the instrument indicates densities from .806 to .843; with No. 1 it registers from .843 to .880, with No. 2 from .880 to .918, with No. 3 from .918 to .958, and with No. 4 from .958 to 1.000, the temperature being 55° F. It will thus be seen that the whole length of the stem corresponds to a difference of density of about .04, and one division to about .00074, indicating a difference of little more than 1/3% in the strength of any sample of spirits.
The instrument is provided with a sliding rule, with scales corresponding to the several weights, which indicate the specific gravity corresponding to the several divisions of the hydrometer scale compared with water at 55° F. The slider upon the rule serves to adjust the scale for different temperatures, and then indicates the strength of the spirit in percentages over or under proof. The slider is also provided with scales, marked respectively Dicas and Clarke, which serve to show the readings which would have been obtained had the instruments of those makers been employed. The line on the scale marked "concentration" indicates the diminution in volume consequent upon reducing the sample to proof strength (if it is _over proof_, O.P.) or upon reducing proof spirit to the strength of the sample (if it is _under proof_, U.P.). By applying the several weights in succession in addition to No. 4 the instrument can be employed for liquids heavier than water; and graduations on the other three sides of the stem, together with an additional slide rule, adapt the instrument for the determination of the strength of worts.
Atkins subsequently modified the instrument (_Nicholson's Journal_, 8vo, iii. 50) by constructing the different weights of different shapes, viz. circular, square, triangular and pentagonal, instead of numbering them 1, 2, 3 and 4 respectively, a figure of the weight being stamped on the sliding rule opposite to every letter in the series to which it belongs, thus diminishing the probability of mistakes. He also replaced the letters on the stem by the corresponding specific gravities referred to water as unity. Further information concerning these instruments and the state of hydrometry in 1803 will be found in Atkins's pamphlet _On the Relation between the Specific Gravities and the Strength of Spirituous Liquors_ (1803); or _Phil. Mag._ xvi. 26-33, 205-212, 305-312; xvii. 204-210 and 329-341.
In Gay-Lussac's alcoholometer the scale is divided into 100 parts corresponding to the presence of 1, 2, ... % by volume of alcohol at 15° C., the highest division of the scale corresponding to the purest alcohol he could obtain (density .7947) and the lowest division corresponding to pure water. A table provides the necessary corrections for other temperatures.
Tralles's hydrometer differs from Gay-Lussac's only in being graduated at 4° C. instead of 15° C., and taking alcohol of density .7939 at 15.5° C. for pure alcohol instead of .7947 as taken by Gay-Lussac (Keene's _Handbook of Hydrometry_).
In Beck's hydrometer the zero of the scale corresponds to density 1.000 and the division 30 to density .850, and equal divisions on the scale are continued as far as is required in both directions.
In the centesimal hydrometer of Francoeur the volume of the stem between successive divisions of the scale is always (1/100)th of the whole volume immersed when the instrument floats in water at 4° C. In order to graduate the stem the instrument is first weighed, then immersed in distilled water at 4° C., and the line of flotation marked zero. The first degree is then found by placing on the top of the stem a weight equal to (1/100)th of the weight of the instrument, which increases the volume immersed by (1/100)th of the original volume. The addition to the top of the stem of successive weights, each (1/100)th of the weight of the instrument itself, serves to determine the successive degrees. The length of 100 divisions of the scale, or the length of the uniform stem the volume of which would be equal to that of the hydrometer up to the zero graduation, Francoeur called the "modulus" of the hydrometer. He constructed his instruments of glass, using different instruments for different portions of the scale (Francoeur, _Traité d'aréométrie_, Paris, 1842).
Dr Boriés of Montpellier constructed a hydrometer which was based upon the results of his experiments on mixtures of alcohol and water. The interval between the points corresponding to pure alcohol and to pure water Boriés divided into 100 equal parts, though the stem was prolonged so as to contain only 10 of these divisions, the other 90 being provided for by the addition of 9 weights to the bottom of the instrument as in Clarke's hydrometer.
The instrument which has now been exclusively used for revenue purposes for nearly a century is that associated with the name of Bartholomew Sikes, who was correspondent to the Board of Excise from 1774 to 1783, and for some time collector of excise for Hertfordshire.
Sikes's hydrometer, on account of its similarity to that of Boriés, appears to have been borrowed from that instrument. It is made of gilded brass or silver, and consists of a spherical ball A (fig. 8), 1.5 in. in diameter, below which is a weight B connected with the ball by a short conical stem C. The stem D is rectangular in section and about 3½ in. in length. This is divided into ten equal parts, each of which is subdivided into five. As in Boriés's instrument, a series of 9 weights, each of the form shown at E, serves to extend the scale to 100 principal divisions. In the centre of each weight is a hole capable of admitting the lowest and thickest end of the conical stem C, and a slot is cut into it just wide enough to allow the upper part of the cone to pass. Each weight can thus be dropped on to the lower stem so as to rest on the counterpoise B. The weights are marked 10, 20, ... 90; and in using the instrument that weight must be selected which will allow it to float in the liquid with a portion only of the stem submerged. Then the reading of the scale at the line of flotation, added to the number on the weight, gives the reading required. A small supernumerary weight F is added, which can be placed upon the top of the stem. F is so adjusted that when the 60 weight is placed on the lower stem the instrument sinks to the same point in distilled water when F is attached as in proof spirit when F is removed. The best instruments are now constructed for revenue purposes of silver, heavily gilded, because it was found that saccharic acid contained in some spirits attacked brass behind the gilding.
The following table gives the specific gravities corresponding to the principal graduations on Sikes's hydrometer at 60° F. and 62° F., together with the corresponding strengths of spirits. The latter are based upon the tables of Charles Gilpin, clerk to the Royal Society, for which the reader is referred to the _Phil. Trans._ for 1794. Gilpin's work is a model for its accuracy and thoroughness of detail, and his results have scarcely been improved upon by more recent workers. The merit of Sikes's system lies not so much in the hydrometer as in the complete system of tables by which the readings of the instrument are at once converted into percentage of proof-spirit.
_Table showing the Densities corresponding to the Indications of Sike's Hydrometer._
+------------+------------------+------------------+ | | 60° F. | 62° F. | | +---------+--------+---------+--------+ | Sike's | Proof | | Proof | | |Indications.| Spirit |Density.| Spirit |Density.| | |per cent.| |per cent.| | +------------+---------+--------+---------+--------+ | 0 | .815297 | 167.0 | .815400 | 166.5 | | 1 | .816956 | 166.1 | .817059 | 165.6 | | 2 | .818621 | 165.3 | .818725 | 164.8 | | 3 | .820294 | 164.5 | .820397 | 163.9 | | 4 | .821973 | 163.6 | .822077 | 163.1 | | 5 | .823659 | 162.7 | .823763 | 162.3 | | 6 | .825352 | 161.8 | .825457 | 161.4 | | 7 | .827052 | 160.9 | .827157 | 160.5 | | 8 | .828759 | 160.0 | .828864 | 159.6 | | 9 | .830473 | 159.1 | .830578 | 158.7 | | 10 | .832195 | 158.2 | .832300 | 157.8 | | 11 | .833888 | 157.3 | .833993 | 156.8 | | 12 | .835587 | 156.4 | .835692 | 155.9 | | 13 | .837294 | 155.5 | .837400 | 155.0 | | 14 | .839008 | 154.6 | .839114 | 154.0 | | 15 | .840729 | 153.7 | .840835 | 153.1 | | 16 | .842458 | 152.7 | .842564 | 152.1 | | 17 | .844193 | 151.7 | .844299 | 151.1 | | 18 | .845936 | 150.7 | .846042 | 150.1 | | 19 | .847685 | 149.7 | .847792 | 149.1 | | 20 | .849442 | 148.7 | .849549 | 148.1 | | 20B | .849393 | 148.7 | .849500 | 148.1 | | 21 | .851122 | 147.6 | .851229 | 147.1 | | 22 | .852857 | 146.6 | .852964 | 146.1 | | 23 | .854599 | 145.6 | .854707 | 145.1 | | 24 | .856348 | 144.6 | .856456 | 144.0 | | 25 | .858105 | 143.5 | .858213 | 142.9 | | 26 | .859869 | 142.4 | .859978 | 141.8 | | 27 | .861640 | 141.3 | .861749 | 140.8 | | 28 | .863419 | 140.2 | .863528 | 139.7 | | 29 | .865204 | 139.1 | .865313 | 138.5 | | 30 | .866998 | 138.0 | .867107 | 137.4 | | 30B | .866991 | 138.0 | .867100 | 137.4 | | 31 | .868755 | 136.9 | .868865 | 136.2 | | 32 | .870526 | 135.7 | .870636 | 135.1 | | 33 | .872305 | 134.5 | .872415 | 133.9 | | 34 | .874090 | 133.4 | .874200 | 132.8 | | 35 | .875883 | 132.2 | .873994 | 131.6 | | 36 | .877684 | 131.0 | .877995 | 130.4 | | 37 | .879492 | 129.8 | .879603 | 129.1 | | 38 | .881307 | 128.5 | .881419 | 127.9 | | 39 | .883129 | 127.3 | .883241 | 126.7 | | 40 | .884960 | 126.0 | .885072 | 125.4 | | 40B | .884888 | 126.0 | .885000 | 125.4 | | 41 | .886689 | 124.8 | .886801 | 124.2 | | 42 | .888497 | 123.5 | .888609 | 122.9 | | 43 | .890312 | 122.2 | .890425 | 121.6 | | 44 | .892135 | 120.9 | .892248 | 120.3 | | 45 | .893965 | 119.6 | .894078 | 119.0 | | 46 | .895803 | 118.3 | .895916 | 117.6 | | 47 | .897647 | 116.9 | .897761 | 116.3 | | 48 | .899509 | 115.6 | .899614 | 114.9 | | 49 | .901360 | 114.2 | .901417 | 113.5 | | 50 | .903229 | 112.8 | .903343 | 112.1 | | 50B | .903186 | 112.8 | .903300 | 112.1 | | 51 | .905024 | 111.4 | .905138 | 110.7 | | 52 | .906869 | 110.0 | .906983 | 109.3 | | 53 | .908722 | 108.6 | .908837 | 107.9 | | 54 | .910582 | 107.1 | .910697 | 106.5 | | 55 | .912450 | 105.6 | .912565 | 105.0 | | 56 | .914326 | 104.2 | .914441 | 103.5 | | 57 | .916209 | 102.7 | .916323 | 102.0 | | 58 | .918100 | 101.3 | .918216 | 100.5 | | 59 | .919999 | 99.7 | .820115 | 98.9 | | 60 | .921906 | 98.1 | .922022 | 97.4 | | 60B | .921884 | 98.1 | .922000 | 97.4 | | 61 | .923760 | 96.6 | .923877 | 95.9 | | 62 | .925643 | 95.0 | .925760 | 94.2 | | 63 | .927534 | 93.3 | .927652 | 92.6 | | 64 | .929433 | 91.7 | .929550 | 90.9 | | 65 | .931339 | 90.0 | .931457 | 89.2 | | 66 | .933254 | 88.3 | .933372 | 87.5 | | 67 | .935176 | 86.5 | .935294 | 85.8 | | 68 | .937107 | 84.7 | .937225 | 84.0 | | 69 | .939045 | 82.9 | .939163 | 82.2 | | 70 | .940991 | 81.1 | .941110 | 80.3 | | 70B | .940981 | 81.1 | .941100 | 80.3 | | 71 | .942897 | 79.2 | .943016 | 78.4 | | 72 | .944819 | 77.3 | .944938 | 76.5 | | 73 | .946749 | 75.3 | .946869 | 74.5 | | 74 | .948687 | 73.3 | .948807 | 72.5 | | 75 | .950634 | 71.2 | .950753 | 70.4 | | 76 | .952588 | 69.0 | .952708 | 68.2 | | 77 | .954550 | 66.8 | .954670 | 66.0 | | 78 | .956520 | 64.4 | .956641 | 63.5 | | 79 | .958498 | 61.9 | .958619 | 61.1 | | 80 | .960485 | 59.4 | .960606 | 58.5 | | 80B | .960479 | 59.4 | .960600 | 58.5 | | 81 | .962433 | 56.7 | .962555 | 55.8 | | 82 | .964395 | 53.9 | .964517 | 53.0 | | 83 | .966366 | 50.9 | .966488 | 50.0 | | 84 | .968344 | 47.8 | .968466 | 47.0 | | 85 | .970331 | 44.5 | .970453 | 43.8 | | 86 | .972325 | 41.0 | .972448 | 40.4 | | 87 | .974328 | 37.5 | .974451 | 36.9 | | 88 | .976340 | 34.0 | .976463 | 33.5 | | 89 | .978359 | 30.6 | .978482 | 30.1 | | 90 | .980386 | 27.2 | .980510 | 26.7 | | 90B | .980376 | 27.2 | .980500 | 26.7 | | 91 | .982371 | 23.9 | .982496 | 23.6 | | 92 | .984374 | 20.8 | .984498 | 20.5 | | 93 | .986385 | 17.7 | .986510 | 17.4 | | 94 | .988404 | 14.8 | .988529 | 14.5 | | 95 | .990431 | 12.0 | .990557 | 11.7 | | 96 | .992468 | 9.3 | .992593 | 9.0 | | 97 | .994512 | 6.7 | .994637 | 6.5 | | 98 | .996565 | 4.1 | .996691 | 4.0 | | 99 | .998626 | 1.8 | .998752 | 1.6 | | 100 |1.000696 | 0.0 |1.000822 | 0.0 | +------------+---------+--------+---------+--------+
In the above table for Sikes's hydrometer two densities are given corresponding to each of the degrees 20, 30, 40, 50, 60, 70, 80 and 90, indicating that the successive weights belonging to the particular instrument for which the table has been calculated do not quite agree. The discrepancy, however, does not produce any sensible error in the strength of the corresponding spirit.
A table which indicates the weight per gallon of spirituous liquors for every degree of Sikes's hydrometer is printed in 23 and 24 Vict. c. 114, schedule B. This table differs slightly from that given above, which has been abridged from the table given in Keene's _Handbook of Hydrometry_, apparently on account of the equal divisions on Sikes's scale having been taken as corresponding to equal increments of density.
Sikes's hydrometer was established for the purpose of collecting the revenue of the United Kingdom by Act of Parliament, 56 Geo. III. c. 140, by which it was enacted that "all spirits shall be deemed and taken to be of the degree of strength which the said hydrometers called Sikes's hydrometers shall, upon trial by any officer or officers of the customs or excise, denote such spirits to be." This act came into force on January 5, 1817, and was to have remained in force until August 1, 1818, but was repealed by 58 Geo. III. c. 28, which established Sikes's hydrometer on a permanent footing. By 3 and 4 Will. IV. c. 52, § 123, it was further enacted that the same instruments and methods should be employed in determining the duty upon imported spirits as should in virtue of any Act of Parliament be employed in the determination of the duty upon spirits distilled at home. It is the practice of the officers of the inland revenue to adjust Sikes's hydrometer at 62° F., that being the temperature at which the imperial gallon is defined as containing 10 lb. avoirdupois of distilled water. The specific gravity of any sample of spirits thus determined, when multiplied by ten, gives the weight in pounds per imperial gallon, and the weight of any bulk of spirits divided by this number gives its volume at once in imperial gallons.
Mr (afterwards Colonel) J. B. Keene, of the Hydrometer Office, London, has constructed an instrument after the model of Sikes's, but provided with twelve weights of different masses but equal volumes, and the instrument is never used without having one of these attached. When loaded with either of the lightest two weights the instrument is specifically lighter than Sikes's hydrometer when unloaded, and it may thus be used for specific gravities as low as that of absolute alcohol. The volume of each weight being the same, the whole volume immersed is always the same when it floats at the same mark whatever weight may be attached.
Besides the above, many hydrometers have been employed for special purposes. Twaddell's hydrometer is adapted for densities greater than that of water. The scale is so arranged that the reading multiplied by 5 and added to 1000 gives the specific gravity with reference to water as 1000. To avoid an inconveniently long stem, different instruments are employed for different parts of the scale as mentioned above.
The lactometer constructed by Dicas of Liverpool is adapted for the determination of the quality of milk. It resembles Sikes's hydrometer in other respects, but is provided with eight weights. It is also provided with a thermometer and slide rule, to reduce the readings to the standard temperature of 55° F. Any determination of density can be taken only as affording prima facie evidence of the quality of milk, as the removal of cream and the addition of water are operations which tend to compensate each other in their influence on the density of the liquid, so that the lactometer cannot be regarded as a reliable instrument.
The marine hydrometers, as supplied by the British government to the royal navy and the merchant marine, are glass instruments with slender stems, and generally serve to indicate specific gravities from 1.000 to 1.040. Before being issued they are compared with a standard instrument, and their errors determined. They are employed for taking observations of the density of sea-water.
The salinometer is a hydrometer originally intended to indicate the strength of the brine in marine boilers in which sea-water is employed. Saunders's salinometer consists of a hydrometer which floats in a chamber through which the water from the boiler is allowed to flow in a gentle stream, at a temperature of 200° F. The peculiarity of the instrument consists in the stream of water, as it enters the hydrometer chamber, being made to impinge against a disk of metal, by which it is broken into drops, thus liberating the steam, which would otherwise disturb the instrument.
The use of Sikes's hydrometer necessitates the employment of a considerable quantity of spirit. For the testing of spirits in bulk no more convenient instrument has been devised, but where very small quantities are available more suitable laboratory methods must be adopted.
In England, the Finance Act 1907 (7 Ed. VII. c. 13), section 4, provides as follows: (1) The Commissioners of Customs and the Commissioners of Inland Revenue may jointly make regulations authorizing the use of any means described in the regulations for ascertaining for any purpose the strength or weight of spirits. (2) Where under any enactment Sykes's (sic) Hydrometer is directed to be used or may be used for the purpose of ascertaining the strength or weight of spirits, any means so authorized by regulations may be used instead of Sykes's Hydrometer and references to Sykes's Hydrometer in any enactment shall be construed accordingly. (3) Any regulations made under this section shall be published in the London, Edinburgh and Dublin Gazette, and shall take effect from the date of publication, or such later date as may be mentioned in the regulations for the purpose. (4) The expression "spirits" in this section has the same meaning as in the Spirits Act 1880. (W. G.)
FOOTNOTES:
[1] In _Nicholson's Journal_, iii. 89, Citizen Eusebe Salverte calls attention to the poem "De Ponderibus et Mensuris" generally ascribed to Rhemnius Fannius Palaemon, and consequently 300 years older than Hypatia, in which the hydrometer is described and attributed to Archimedes.
[2] _Nicholson's Journal_, vol. i. p. 111, footnote.
HYDROPATHY, the name given, from the Greek, to the "water-cure," or the treatment of disease by water, used outwardly and inwardly. Like many descriptive names, the word "hydropathy" is defective and even misleading, the active agents in the treatment being heat and cold, of which water is little more than the vehicle, and not the only one. Thermotherapeutics (or thermotherapy) is a term less open to objection.
Hydropathy, as a formal system, dates from about 1829, when Vincenz Priessnitz (1801-1851), a farmer of Gräfenberg in Silesia, Austria, began his public career in the paternal homestead, extended so as to accommodate the increasing numbers attracted by the fame of his cures. Two English works, however, on the medical uses of water had been translated into German in the century preceding the rise of the movement under Priessnitz. One of these was by Sir John Floyer (1649-1734), a physician of Lichfield, who, struck by the remedial use of certain springs by the neighbouring peasantry, investigated the history of cold bathing, and published in 1702 his [Greek: "Psychrolousia], _or the History of Cold Bathing, both Ancient and Modern_." The book ran through six editions within a few years, and the translation was largely drawn upon by Dr J. S. Hahn of Silesia, in a work published in 1738, _On the Healing Virtues of Cold Water, Inwardly and Outwardly applied, as proved by Experience_. The other work was that of Dr James Currie (1756-1805) of Liverpool, entitled _Medical Reports on the Effects of Water, Cold and Warm, as a remedy in Fevers and other Diseases_, published in 1797, and soon after translated into German by Michaelis (1801) and Hegewisch (1807). It was highly popular, and first placed the subject on a scientific basis. Hahn's writings had meanwhile created much enthusiasm among his countrymen, societies having been everywhere formed to promote the medicinal and dietetic use of water; and in 1804 Professor Örtel of Ansbach republished them and quickened the popular movement by unqualified commendation of water drinking as a remedy for all diseases. In him the rising Priessnitz found a zealous advocate, and doubtless an instructor also.
At Gräfenberg, to which the fame of Priessnitz drew people of every rank and many countries, medical men were conspicuous by their numbers, some being attracted by curiosity, others by the desire of knowledge, but the majority by the hope of cure for ailments which had as yet proved incurable. Many records of experiences at Gräfenberg were published, all more or less favourable to the claims of Priessnitz, and some enthusiastic in their estimate of his genius and penetration; Captain Claridge introduced hydropathy into England in 1840, his writings and lectures, and later those of Sir W. Erasmus Wilson (1809-1884), James Manby Gully (1808-1883) and Edward Johnson, making numerous converts, and filling the establishments opened soon after at Malvern and elsewhere. In Germany, France and America hydropathic establishments multiplied with great rapidity. Antagonism ran high between the old practice and the new. Unsparing condemnation was heaped by each on the other; and a legal prosecution, leading to a royal commission of inquiry, served but to make Priessnitz and his system stand higher in public estimation.
Increasing popularity diminished before long that timidity which had in great measure prevented trial of the new method from being made on the weaker and more serious class of cases, and had caused hydropathists to occupy themselves mainly with a sturdy order of chronic invalids well able to bear a rigorous regimen and the severities of unrestricted crisis. The need of a radical adaptation to the former class was first adequately recognized by John Smedley, a manufacturer of Derbyshire, who, impressed in his own person with the severities as well as the benefits of "the cold water cure," practised among his workpeople a milder form of hydropathy, and began about 1852 a new era in its history, founding at Matlock a counterpart of the establishment at Gräfenberg.
Ernst Brand (1826-1897) of Berlin, Räljen and Theodor von Jürgensen of Kiel, and Karl Liebermeister (1833-1901) of Basel, between 1860 and 1870, employed the cooling bath in abdominal typhus with striking results, and led to its introduction to England by Dr Wilson Fox. In the Franco-German war the cooling bath was largely employed, in conjunction frequently with quinine; and it now holds a recognized position in the treatment of hyperpyrexia. The wet sheet pack has become part of medical practice; the Turkish bath, introduced by David Urquhart (1805-1877) into England on his return from the East, and ardently adopted by Dr Richard Barter (1802-1870) of Cork, has become a public institution, and, with the "morning tub" and the general practice of water drinking, is the most noteworthy of the many contributions by hydropathy to public health (see BATHS, ad fin.).
The appliances and arrangements by means of which heat and cold are brought to bear on the economy are--(a) Packings, hot and cold, general and local, sweating and cooling; (b) hot air and steam baths; (c) general baths, of hot water and cold; (d) sitz, spinal, head and foot baths; (e) bandages (or compresses), wet and dry; also (f) fomentations and poultices, hot and cold, sinapisms, stupes, rubbings and water potations, hot and cold.
(a) Packings.--The full pack consists of a wet sheet enveloping the body, with a number of dry blankets packed tightly over it, including a macintosh covering or not. In an hour or less these are removed and a general bath administered. The pack is a derivative, sedative, sudorific and stimulator of cutaneous excretion. There are numerous modifications of it, notably the cooling pack, where the wrappings are loose and scanty, permitting evaporation, and the application of indefinite duration, the sheet being rewetted as it dries; this is of great value in protracted febrile conditions. There are also local packs, to trunk, limbs or head separately, which are derivative, soothing or stimulating, according to circumstance and detail.
(b) Hot air baths, the chief of which is the Turkish (properly, the Roman) bath, consisting of two or more chambers ranging in temperature from 120° to 212° or higher, but mainly used at 150° for curative purposes. Exposure is from twenty minutes up to two hours according to the effect sought, and is followed by a general bath, and occasionally by soaping and shampooing. It is stimulating, derivative, depurative, sudorific and alterative, powerfully promoting tissue change by increase of the natural waste and repair. It determines the blood to the surface, reducing internal congestions, is a potent diaphoretic, and, through the extremes of heat and cold, is an effective nervous and vascular stimulant and tonic. Morbid growths and secretions, as also the uraemic, gouty and rheumatic diathesis, are beneficially influenced by it. The full pack and Turkish bath have between them usurped the place and bettered the function of the once familiar hot bath. The Russian or steam bath and the lamp bath are primitive and inferior varieties of the modern Turkish bath, the atmosphere of which cannot be too dry and pure.
(c) General baths comprise the rain (or needle), spray (or rose), shower, shallow, plunge, douche, wave and common morning sponge baths, with the dripping sheet, and hot and cold spongings, and are combinations, as a rule, of hot and cold water. They are stimulating, tonic, derivative and detergent.
(d) Local baths comprise the sitz (or sitting), douche (or spouting), spinal, foot and head baths, of hot or cold water, singly or in combination, successive or alternate. The sitz, head and foot baths are used "flowing" on occasion. The application of cold by "Leiter's tubes" is effective for reducing inflammation (e.g. in meningitis and in sunstroke); in these a network of metal or indiarubber tubing is fitted to the part affected, and cold water kept continuously flowing through them. Rapid alternations of hot and cold water have a powerful effect in vascular stasis and lethargy of the nervous system and absorbents, yielding valuable results in local congestions and chronic inflammations.
(e) Bandages (or compresses) are of two kinds,--_cooling_, of wet material left exposed for evaporation, used in local inflammations and fevers; and _heating_, of the same, covered with waterproof material, used in congestion, external or internal, for short or long periods. Poultices, warm, of bread, linseed, bran, &c., changed but twice in twenty-four hours, are identical in action with the heating bandage, and superior only in the greater warmth and consequent vital activity their closer application to the skin ensures.
(f) Fomentations and poultices, hot or cold, sinapisms, stupes, rubefacients, irritants, frictions, kneadings, calisthenics, gymnastics, electricity, &c., are adjuncts largely employed.
BIBLIOGRAPHY.--Among the numerous earlier works on hydropathy, the following are worth mention: Balbirnie, _Water Cure in Consumption_ (1847), _Hydropathic Aphorisms_ (1856) and _A Plea for the Turkish Bath_ (1862); Beni-Barde, _Traité d'hydrothérapie_ (1874); Claridge, _Cold Water Cure, or Hydropathy_ (1841), _Facts and Evidence in Support of Hydropathy_ (1843) and _Cold Water, Tepid Water and Friction Cure_ (1849); Dunlop, _Philosophy of the Bath_ (1873); Floyer, _Psychrolousia, or the History of Cold-Bathing_, &c. (1702); J. S. Hahn (Schweidnitz), _Observations on the Healing Virtues of Cold Water_ (1738); Hunter, _Hydropathy for Home Use_ (1879); E. W. Lane, _Hydropathy, or the Natural System of Medical Treatment_ (1857); R. J. Lane, _Life at the Water Cure_ (1851); Shew, _Hydropathic Family Physician_ (1857); Smedley, _Practical Hydropathy_ (1879); Smethurst, _Hydrotherapia, or the Water Cure_ (1843); Wainwright, _Inquiry into the Nature and Use of Baths_ (1737); Weiss, _Handbook of Hydropathy_ (1844); Wilson _Principles and Practice of the Cold Water Cure_ (1854) and _The Water Cure_ (1859). A useful recent work dealing comprehensively with the subject is Richard Metcalfe's _Rise and Progress of Hydropathy_ (1906).
HYDROPHOBIA (Gr. [Greek: Hydor], water, and [Greek: phobos], fear; so called from the symptom of dread of water), or RABIES (Lat. for "madness"), an acute disease, occurring chiefly in certain of the lower animals, particularly the canine species, and liable to be communicated by them to other animals and to man.
_In Dogs, &c._--The occurrence of rabies in the fox, wolf, hyaena, jackal, raccoon, badger and skunk has been asserted; but there is every probability that it is originally a disease of the dog. It is communicated by inoculation to nearly all, if not all, warm-blooded creatures. The transmission from one animal to another only certainly takes place through inoculation with viruliferous matters. The malady is generally characterized at a certain stage by an irrepressible desire in the animal to act offensively with its natural weapons--dogs and other carnivora attacking with their teeth, herbivora with their hoofs or horns, and birds with their beaks, when excited ever so slightly. In the absence of excitement the malady may run its course without any fit of fury or madness.
_Symptoms._--The disease has been divided into three stages or periods, and has also been described as appearing in at least two forms, according to the peculiarities of the symptoms. But, as a rule, one period of the disease does not pass suddenly into another, the transition being almost imperceptible; and the forms do not differ essentially from each other, but appear merely to constitute varieties of the same disease, due to the natural disposition of the animal, or other modifying circumstances. These forms have been designated _true_ or _furious rabies_ (Fr. _rage vrai_; Ger. _rasende Wuth_) and _dumb rabies_ (Fr. _rage mue_; Ger. _stille Wuth_).
The malady does not commence with fury and madness, but in a strange and anomalous change in the habits of the dog: it becomes dull, gloomy, and taciturn, and seeks to isolate itself in out-of-the-way places, retiring beneath chairs and to odd corners. But in its retirement it cannot rest: it is uneasy and fidgety, and no sooner has it lain down than suddenly it jumps up in an agitated manner, walks backwards and forwards several times, again lies down and assumes a sleeping attitude, but has only maintained it for a few minutes when it is once more moving about. Again it retires to its corner, to the farthest recess it can find, and huddles itself up into a heap, with its head concealed beneath its chest and fore-paws. This state of continual agitation and inquietude is in striking contrast with its ordinary habits, and should therefore receive attention. Not unfrequently there are a few moments when the creature appears more lively than usual, and displays an extraordinary amount of affection. Sometimes there is a disposition to gather up straw, thread, bits of wood, &c., which are industriously carried away; a tendency to lick anything cold, as iron, stones, &c., is also observed in many instances; and there is also a desire evinced to lick other animals. Sexual excitement is also frequently an early symptom. At this period no disposition to bite is observed; the animal is docile with its master and obeys his voice, though not so readily as before, nor with the same pleased countenance. There is something strange in the expression of its face, and the voice of its owner is scarcely able to make it change from a sudden gloominess to its usual animated aspect. These symptoms gradually become more marked; the restlessness and agitation increase. If on straw the dog scatters and pulls it about with its paws, and if in a room it scratches and tumbles the cushions or rugs on which it usually lies. It is incessantly on the move, rambling about, scratching the ground, sniffing in corners and at the doors, as if on the scent or seeking for something. It indulges in strange movements, as if affected by some mental influences or a prey to hallucinations. When not excited by any external influence it will remain for a brief period perfectly still and attentive, as if watching something, or following the movements of some creature on the wall; then it will suddenly dart forward and snap at the vacant air, as if pursuing an annoying object, or endeavouring to seize a fly. At another time it throws itself, yelling and furious, against the wall, as if it heard threatening voices on the other side, or was bent on attacking an enemy. Nevertheless, the animal is still docile and submissive, for its master's voice will bring it out of its frenzy. But the saliva is already virulent, and the excessive affection which it evinces at intervals, by licking the hands or face of those it loves, renders the danger very great should there be a wound or abrasion. Until a late period in the disease the master's voice has a powerful influence over the animal. When it has escaped from all control and wanders erratically abroad, ferocious and restless, and haunted by horrid phantoms, the familiar voice yet exerts its influence, and it is rare indeed that it attacks its master.
There is no dread of water in the rabid dog; the animal is generally thirsty, and if water be offered will lap it with avidity, and swallow it at the commencement of the disease. And when, at a later period, the constriction about the throat--symptomatic of the disease--renders swallowing difficult, the dog will none the less endeavour to drink, and the lappings are as frequent and prolonged when deglutition becomes impossible. So little dread has the rabid dog of water that it will ford streams and swim rivers; and when in the ferocious stage it will even do this in order to attack other creatures on the opposite side.
At the commencement of the disease the dog does not usually refuse to eat, and some animals are voracious to an unusual degree. But in a short time it becomes fastidious, only eating what it usually has a special predilection for. Soon, however, this gives place to a most characteristic symptom--either the taste becomes extremely depraved or the dog has a fatal and imperious desire to bite and ingest everything. The litter of its kennel, wool from cushions, carpets, stockings, slippers, wood, grass, earth, stones, glass, horse-dung, even its own faeces and urine, or whatever else may come in its way, are devoured. On examination of the body of a dog which has died of rabies it is so common to find in the stomach a quantity of dissimilar and strange matters on which the teeth have been exercised that, if there was nothing known of the animal's history, there would be strong evidence of its having been affected with the disease. When a dog, then, is observed to gnaw and eat suchlike matters, though it exhibits no tendency to bite, it should be suspected.
The mad dog does not usually foam at the mouth to any great extent at first. The mucus of the mouth is not much increased in quantity, but it soon becomes thicker, viscid, and glutinous, and adheres to the angles of the mouth, fauces and teeth. It is at this period that the thirst is most ardent, and the dog sometimes furiously attempts to detach the saliva with its paws; and if after a while it loses its balance in these attempts and tumbles over, there can no longer be any doubt as to the nature of the malady. There is another symptom connected with the mouth in that form of the disease named "dumb madness" which has frequently proved deceptive. The lower jaw drops in consequence of paralysis of its muscles, and the mouth remains open. The interior is dry from the air passing continually over it, and assumes a deep red tint, somewhat masked by patches of dust or earth, which more especially adhere to the upper surface of the tongue and to the lips. The strange alteration produced in the dog's physiognomy by its constantly open mouth and the dark colour of the interior is rendered still more characteristic by the dull, sad, or dead expression of the animal's eyes. In this condition the creature is not very dangerous, because generally it could not bite if it tried--indeed there does not appear to be much desire to bite in dumb madness; but the saliva is none the less virulent, and accidental inoculations with it, through imprudent handling, will prove as fatal as in the furious form. The mouth should not be touched,--numerous deaths having occurred through people thinking the dog had some foreign substance lodged in its throat, and thrusting their fingers down to remove it. The sensation of tightness which seems to exist at the throat causes the dog to act as if a bone were fixed between its teeth or towards the back of its mouth, and to employ its fore-paws as if to dislodge it. This is a very deceptive symptom, and may prove equally dangerous if caution be not observed. Vomiting of blood or a chocolate-coloured fluid is witnessed in some cases, and has been supposed to be due to the foreign substances in the stomach, which abrade the lining membrane; this, however, is not correct, as it has been observed in man.
The voice of the rabid dog is very peculiar, and so characteristic that to those acquainted with it nothing more is needed to prove the presence of the disease. Those who have heard it once or twice never forget its signification. Owing to the alterations taking place in the larynx the voice becomes hoarse, cracked and stridulous, like that of a child affected with croup--the "voix du coq," as the French have it. A preliminary bark is made in a somewhat elevated tone and with open mouth; this is immediately succeeded by five, six or eight decreasing howls, emitted when the animal is sitting or standing, and always with the nose elevated, which seem to come from the depths of the throat, the jaws not coming together and closing the mouth during such emission, as in the healthy bark. This alteration in the voice is frequently the first observable indication of the malady, and should at once attract attention. In dumb madness the voice is frequently lost from the very commencement--hence the designation.
The sensibility of the mad dog appears to be considerably diminished, and the animal appears to have lost the faculty of expressing the sensations it experiences: it is mute under the infliction of pain, though there can be no doubt that it still has peripheral sensation to some extent. Burning, beating and wounding produce much less effect than in health, and the animal will even mutilate itself with its teeth. Suspicion, therefore, should always strongly attach to a dog which does not manifest a certain susceptibility to painful impressions and receives punishment without any cry or complaint. There is also reason for apprehension when a dog bites itself persistently in any part of its body. A rabid dog is usually stirred to fury at the sight of one of its own species; this test has been resorted to by Henrie Marie Bouley (1814-1885) to dissipate doubts as to the existence of the disease when the diagnosis is otherwise uncertain. As soon as the suspected animal, if it is really rabid, finds itself in the presence of another of its species it at once assumes the aggressive, and, if allowed, will bite furiously. All rabid animals indeed become excited, exasperated, and furious at the sight of a dog, and attack it with their natural weapons, even the timid sheep when rabid butts furiously at the enemy before which in health it would have fled in terror. This inversion of sentiment is sometimes valuable in diagnosing the malady; it is so common that it may be said to be present in every case of rabies. When, therefore, a dog, contrary to its habits and natural inclination, becomes suddenly aggressive to other dogs, it is time to take precautions.
In the large majority of instances the dog is inoffensive in the early period of the disease to those to whom it is familiar. It then flies from its home and either dies, is killed as "mad," or returns in a miserable plight, and in an advanced stage of the malady, when the desire to bite is irresistible. It is in the early stage that sequestration and suppressive measures are most valuable. The dogs which propagate the disease are usually those that have escaped from their owners. After two or three days, frequently in about twelve hours, more serious and alarming symptoms appear, ferocious instincts are developed, and the desire to do injury is irrepressible. The animal has an indefinable expression of sombre melancholy and cruelty. The eyes have their pupils dilated, and emit flashes of light when they are not dull and heavy; they always appear so fierce as to produce terror in the beholder; they are red, and their sensibility to light is increased; and wrinkles, which sometimes appear on the forehead, add to the repulsive aspect of the animal. If caged it flies at the spectator, emitting its characteristic howl or bark, and seizing the iron bars with its teeth, and if a stick be thrust before it this is grasped and gnawed. This fury is soon succeeded by lassitude, when the animal remains insensible to every excitement. Then all at once it rouses up again, and another paroxysm of fury commences. The first paroxysm is usually the most intense, and the fits vary in duration from some hours to a day, and even longer; they are ordinarily briefer in trained and pet dogs than in those which are less domesticated, but in all the remission is so complete after the first paroxysm that the animals appear to be almost well, if not in perfect health. During the paroxysms respiration is hurried and laboured, but tranquil during the remissions. There is an increase of temperature, and the pulse is quick and hard. When the animal is kept in a dark place and not excited, the fits of fury are not observed. Sometimes it is agitated and restless in the manner already described. It never becomes really furious or aggressive unless excited by external objects--the most potent of these, as has been said, being another dog, which, however, if it be admitted to its cage, it may not at once attack. The attacked animal rarely retaliates, but usually responds to the bites by acute yells, which contrast strangely with the silent anger of the aggressor, and tries to hide its head with its paws or beneath the straw. These repeated paroxysms hurry the course of the disease. The secretion and flowing of a large quantity of saliva from the mouth are usually only witnessed in cases in which swallowing has become impossible, the mouth being generally dry. At times the tongue, nose and whole head appear swollen. Other dogs frequently shun one which is rabid, as if aware of their danger.
The rabid dog, if lodged in a room or kept in a house, is continually endeavouring to escape; and when it makes its escape it goes freely forward, as if impelled by some irresistible force. It travels considerable distances in a short time, perhaps attacking every living creature it meets--preferring dogs, however, to other animals, and these to mankind; cats, sheep, cattle and horses are particularly liable to be injured. It attacks in silence, and never utters a snarl or a cry of anger; should it chance to be hurt in return it emits no cry or howl of pain. The degree of ferocity appears to be related to natural disposition and training. Some dogs, for instance, will only snap or give a slight bite in passing, while others will bite furiously, tearing the objects presented to them, or which they meet in their way, and sometimes with such violence as to injure their mouth and break their teeth, or even their jaws. If chained, they will in some cases gnaw the chain until their teeth are worn away and the bones laid bare. The rabid dog does not continue its progress very long. Exhausted by fatigue and the paroxysms of madness excited in it by the objects it meets, as well as by hunger, thirst, and also, no doubt, by the malady, its limbs soon become feeble; the rate of travelling is lessened and the walk is unsteady, while its drooping tall, head inclined towards the ground, open mouth, and protruded tongue (of a leaden colour or covered with dust) give the distressed creature a very striking and characteristic physiognomy. In this condition, however, it is much less to be dreaded than in its early fits of fury, since it is no longer capable or desirous of altering its course or going out of its way to attack an animal or a man not immediately in the path. It is very probable that its fast-failing vision, deadened scent, and generally diminished perception prevent its being so readily impressed or excited by surrounding objects as it previously was. To each paroxysm, which is always of short duration, there succeeds a degree of exhaustion as great as the fits have been violent and oft repeated. This compels the animal to stop; then it shelters itself in obscure places--frequently in ditches by the roadside--and lies there in a somnolescent state for perhaps hours. There is great danger, nevertheless, in disturbing the dog at this period; for when roused from its torpor it has sometimes sufficient strength to inflict a bite. This period, which may be termed the second stage, is as variable in its duration as the first, but it rarely exceeds three or four days. The above-described phenomena gradually merge into those of the third or last period, when symptoms of paralysis appear, which are speedily followed by death. During the remission in the paroxysms these paralytic symptoms are more particularly manifested in the hind limbs, which appear as if unable to support the animal's weight, and cause it to stagger about; or the lower jaw becomes more or less drooping, leaving the parched mouth partially open. Emaciation rapidly sets in, and the paroxysms diminish in intensity, while the remissions become less marked. The physiognomy assumes a still more sinister and repulsive aspect; the hair is dull and erect; the flanks are retracted; the eyes lose their lustre and are buried in the orbits, the pupil being dilated, and the cornea dull and semi-opaque; very often, even at an early period, the eyes squint, and this adds still more to the terrifying appearance of the poor dog. The voice, if at all heard, is husky, the breathing laborious, and the pulse hurried and irregular. Gradually the paralysis increases, and the posterior extremities are dragged as if the animal's back were broken, until at length it becomes general; it is then the prelude to death. Or the dog remains lying in a state of stupor, and can only raise itself with difficulty on the fore-limbs when greatly excited. In this condition it may yet endeavour to bite at objects within its reach. At times convulsions of a tetanic character appear in certain muscles; at other times these are general. A comatose condition ensues, and the rabid dog, if permitted to die naturally, perishes, in the great majority of cases, from paralysis and asphyxia.
In dumb madness there is paralysis of the lower jaw, which imparts a curious and very characteristic physiognomy to the dog; the voice is also lost, and the animal can neither eat nor drink. In this condition the creature remains with its jaw pendent and the mouth consequently wide open, showing the flaccid or swollen tongue covered with brownish matter, and a stringy gelatinous-looking saliva lying between it and the lower lip and coating the fauces, which sometimes appear to be inflamed. Though the animal is unable to swallow fluids, the desire to drink is nevertheless intense; for the creature will thrust its face into the vessel of water in futile attempts to obtain relief, even until the approach of death. Water may be poured down its throat without inducing a paroxysm. The general physiognomy and demeanour of the poor creature inspire the beholder with pity rather than fear. The symptoms due to cerebral excitement are less marked than in the furious form of the disease; the agitation is not so considerable, and the restlessness, tendency to run away, and desire to bite are nearly absent; generally the animal is quite passive. Not unfrequently one or both eyes squint, and it is only when very much excited that the dog may contrive to close its mouth. Sometimes there is swelling about the pharynx and the neck; when the tongue shares in this complication it hangs out of the mouth. In certain cases there is a catarrhal condition of the membrane lining the nasal cavities, larynx, and bronchi; sometimes the animal testifies to the existence of abdominal pain, and the faeces are then soft or fluid. The other symptoms--such as the rapid exhaustion and emaciation, paralysis of the posterior limbs towards the termination of the disease, as well as the rapidity with which it runs its course--are the same as in the furious form.
The simultaneous occurrence of furious and dumb madness has frequently been observed in packs of fox-hounds. Dumb madness differs, then, from the furious type in the paralysis of the lower jaw, which hinders the dog from biting, save in very exceptional circumstances; the ferocious instincts are also in abeyance; and there is no tendency to aggression. It has been calculated that from 15 to 20% of rabid dogs have this particular form of the disease. Puppies and young dogs chiefly have furious rabies.
These are the symptoms of rabies in the dog; but it is not likely, nor is it necessary, that they will all be present in every case. In other species the symptoms differ more or less from those manifested by the dog, but they are generally marked by a change in the manner and habits of the creatures affected, with strong indications of nervous disturbance, in the majority of species amounting to ferociousness and a desire to injure, timid creatures becoming bold and aggressive.
_In Human Beings._--The disease of hydrophobia has been known from early times, and is alluded to in the works of Aristotle, Xenophon, Plutarch, Virgil, Horace, Ovid and many others, as well as in those of the early writers on medicine. Celsus gives detailed instructions respecting the treatment of men who have been bitten by rabid dogs, and dwells on the dangers attending such wounds. After recommending suction of the bitten part by means of a dry cupping glass, and thereafter the application of the actual cautery or of strong caustics, and the employment of baths and various internal remedies, he says: "Idque cum ita per triduum factum est, tutus esse homo a periculo videtur. Solet autem ex eo vulnere, ubi parum occursum est, aquae timor nasci, [Greek: hydrophobian] Graeci appellant. Miserrimum genus morbi; in quo simul aeger et siti et aquae metu cruciatur; quo oppressis in angusto spes est." Subsequently Galen described minutely the phenomena of hydrophobia, and recommended the excision of the wounded part as a protection against the disease. Throughout many succeeding centuries little or nothing was added to the facts which the early physicians had made known upon the subject. The malady was regarded with universal horror and dread, and the unfortunate sufferers were generally abandoned by all around them and left to their terrible fate. In later times the investigations of Boerhaave, Gerard van Swieten (1700-1772), John Hunter, François Magendie (1783-1855), Gilbert Breschet (1784-1845), Virchow, Albert Reder, as also of William Youatt (1776-1847), George Fleming, Meynell, Karl Hertwig (1798-1881), and others, have furnished important information; but all these were put into the shade by the researches of Pasteur.
The disease is communicated by the secretions of the mouth of the affected animal entering a wound or abrasion of the human skin or mucous membrane. In the great majority of cases (90%) this is due to the bite of a rabid dog, but bites of rabid cats, wolves, foxes, jackals, &c. are occasionally the means of conveying the disease. Numerous popular fallacies still prevail on the subject of hydrophobia. Thus it is supposed that the bite of an angry dog may produce the disease, and all the more if the animal should subsequently develop symptoms of rabies. The ground for this erroneous notion is the fact, which is unquestionable, that animals in whom rabies is in the stage of incubation, during which there are few if any symptoms, may by their bites convey the disease, though fortunately during this early stage they are little disposed to bite. The bite of a non-rabid animal, however enraged, cannot give rise to hydrophobia.
The period of incubation of the disease, or that time which elapses between the introduction of the virus and the development of the symptoms, appears to vary in a remarkable degree, being in some cases as short as a fortnight, and in others as long as several months or even years. On an average it seems to be from about six weeks to three months, but it mainly depends on the part bitten; bites on the head are the most dangerous. The incubation period is also said to be shorter in children. The rare instances of the appearance of hydrophobia many years after the introduction of the poison are always more or less open to question as to subsequent inoculation.
When the disease is about to declare itself it not unfrequently happens that the wound, which had quickly and entirely healed after the bite, begins to exhibit evidence of irritation or inflammatory action, or at least to be the seat of morbid sensations such as numbness, tingling or itching. The symptoms characterizing the premonitory stage are great mental depression and disquietude, together with restlessness and a kind of indefinite fear. There is an unusual tendency to talk, and the articulation is abrupt and rapid. Although in some instances the patients will not acknowledge that they have been previously bitten, and deny it with great obstinacy, yet generally they are well aware of the nature of their malady, and speak despairingly of its consequences. There is in this early stage a certain amount of constitutional disturbance showing itself by feverishness, loss of appetite, sleeplessness, headache, great nervous excitability, respiration of a peculiar sighing or sobbing character, and even occasionally a noticeable aversion to liquids. These symptoms--constituting what is termed the melancholic stage--continue in general for one or two days, when they are succeeded by the stage of excitement in which all the characteristic phenomena of the malady are fully developed. Sometimes the disease first shows itself in this stage, without antecedent symptoms.
The agitation of the sufferer now becomes greatly increased, and the countenance exhibits anxiety and terror. There is noticed a marked embarrassment of the breathing, but the most striking and terrible features of this stage are the effects produced by attempts to swallow fluids. The patient suffers from thirst and desires eagerly to drink, but on making the effort is seized with a most violent suffocative paroxysm produced by spasm of the muscles of swallowing and breathing, which continues for several seconds, and is succeeded by a feeling of intense alarm and distress. With great caution and determination the attempt is renewed, but only to be followed with a repetition of the seizure, until the unhappy sufferer ceases from sheer dread to try to quench the thirst which torments him. Indeed the very thought of doing so suffices to bring on a choking paroxysm, as does also the sound of the running of water. The patient is extremely sensitive to any kind of external impression; a bright light, a loud noise, a breath of cool air, contact with any one, are all apt to bring on one of these seizures. But besides these suffocative attacks there also occur general convulsions affecting the whole muscular system of the body, and occasionally a condition of tetanic spasm. These various paroxysms increase in frequency and severity with the advance of the disease, but alternate with intervals of comparative quiet, in which, however, there is intense anxiety and more or less constant difficulty of breathing, accompanied with a peculiar sonorous expiration, which has suggested the notion that the patient barks like a dog. In many instances there is great mental disturbance, with fits of maniacal excitement, in which he strikes at every one about him, and accuses them of being the cause of his sufferings--these attacks being succeeded by calm intervals in which he expresses great regret for his violent behaviour. During all this stage of the disease the patient is tormented with a viscid secretion accumulating in his mouth, which from dread of swallowing he is constantly spitting about him. There may also be noticed snapping movements of the jaws as if he were attempting to bite, but these are in reality a manifestation of the spasmodic action which affects the muscles generally. There is no great amount of fever, but there is constipation, diminished flow of urine, and often sexual excitement.
After two or three days of suffering of the most terrible description the patient succumbs, death taking place either in a paroxysm of choking, or on the other hand in a tranquil manner from exhaustion, all the symptoms having abated, and the power of swallowing returned before the end. The duration of the disease from the first declaration of the symptoms is generally from three to five days.
Apart from the inoculation method (see below), the treatment of most avail is that which is directed towards preventing the absorption of the poison into the system. This may be accomplished by excision of the part involved in the bite of the rabid animal, or, where this from its locality is impracticable, in the application to the wound of some chemical agent which will destroy the activity of the virus, such as potassa fusa, lunar caustic (nitrate of silver), or the actual cautery in the form of a red-hot wire. The part should be thoroughly acted on by these agents, no matter what amount of temporary suffering this may occasion. Such applications should be resorted to immediately after the bite has been inflicted, or as soon thereafter as possible. Further, even though many hours or days should elapse, these local remedies should still be applied; for if, as appears probable, some at least of the virus remains for long at the injured part, the removal or effectual destruction of this may prevent the dread consequences of its absorption. Every effort should be made to tranquillize and reassure the patient.
Two special points of interest have arisen in recent years in connexion with this disease. One is the Pasteur treatment by inoculation with rabic virus (see also PARASITIC DISEASES), and the other was the attempt of the government to exterminate rabies in the British Isles by muzzling dogs.
Pasteur treatment.
The Pasteur treatment was first applied to human beings in 1885 after prolonged investigation and experimental trial on animals. It is based on the fact that a virus, capable of giving rabies by inoculation, can be extracted from the tissues of a rabid animal and then intensified or attenuated at pleasure. It appears that the strength of the rabic virus, as determined by inoculation, is constant in the same species of animal, but is modified by passing through another species. For instance, the natural virus of dogs is always of the same strength, but when inoculated into monkeys it becomes weakened, and the process of attenuation can be carried on by passing the virus through a succession of monkeys, until it loses the power of causing death. If this weakened virus is then passed back through guinea-pigs, dogs or rabbits, it regains its former strength. Again, if it be passed through a succession of dogs it becomes intensified up to a maximum of strength which is called the _virus fixe_. Pasteur further discovered that the strength can be modified by temperature and by keeping the dried tissues of a rabid animal containing the virus. Thus, if the spinal cord of a rabid dog be preserved in a dry state, the virus loses strength day by day. The system of treatment consists in making an emulsion of the cord and graduating the strength of the dose by using a succession of cords, which have been kept for a progressively diminishing length of time. Those which have been kept for fourteen days are used as a starting-point, yielding virus of a minimum strength. They are followed by preparations of diminishing age and increasing strength, day by day, up to the maximum, which is three days old. These are successively injected into the circulatory system. The principle is the artificial acquisition by the patient of resistance to the rabic virus, which is presumed to be already in the system but has not yet become active, by accustoming him gradually to its toxic effect, beginning with a weak form and progressively increasing the dose. It is not exactly treatment of the disease, because it is useless or nearly so when the disease has commenced, nor is it exactly preventive, for the patient has already been bitten. It must be regarded as a kind of anticipatory cure. The cords are cut into sections and preserved dry in sterilized flasks plugged with cotton-wool. Another method of preparing the inoculatory virus, which has been devised by Guido Tizzoni and Eugenio Centanni, consists in subjecting the _virus fixe_ to peptic digestion by diluted gastric juice for varying periods of time.
The first patient was treated by Pasteur's system in July 1885. He was successively inoculated with emulsions made from cords that had been kept fourteen and ten days, then eleven and eight days, then eight, seven, six days, and so on. Two forms of treatment are now used--(1) the "simple," in which the course from weak to strong virus is extended over nine days; (2) the "intensive," in which the maximum is reached in seven days. The latter is used in cases of very bad bites and those of some standing, in which it is desirable to lose no time. Two days are compressed into one at the commencement by making injections morning and evening instead of once a day, so that the fifth-day cord is reached in four days instead of six, as in the "simple" treatment. When the maximum--the third-day cord--is reached the injections are continued with fifth-, fourth-, and third-day cords. The whole course is fifteen days in the simple treatment and twenty-one in the intensive. The doses injected range from 1 to 3 cubic centimetres. Injections are made alternately into the right and left flanks. The following table shows the number treated from 1886 to 1905, with the mortality.
+------+--------+-------+---------+ | Year.|Patients|Deaths.|Mortality| | |Treated.| |per cent.| +------+--------+-------+---------+ | 1886 | 2671 | 25 | .94 | | 1887 | 1770 | 14 | .79 | | 1888 | 1622 | 9 | .55 | | 1889 | 1830 | 7 | .38 | | 1890 | 1540 | 5 | .32 | | 1891 | 1559 | 4 | .25 | | 1892 | 1790 | 4 | .22 | | 1893 | 1648 | 6 | .36 | | 1894 | 1387 | 7 | .50 | | 1895 | 1520 | 5 | .33 | | 1896 | 1308 | 4 | .30 | | 1897 | 1521 | 6 | .39 | | 1898 | 1465 | 3 | .20 | | 1899 | 1614 | 4 | .25 | | 1900 | 1419 | 10 | .70 | | 1901 | 1318 | 5 | .37 | | 1902 | 1105 | 2 | .18 | | 1903 | 630 | 4 | .65 | | 1904 | 757 | 5 | .66 | | 1905 | 727 | 4 | .54 | +------+--------+-------+---------+
These figures do not include cases which develop hydrophobia during treatment or within fifteen days after treatment is completed, for it is held that persons who die within that period have their nervous centres invaded by virus before the cure has time to act. The true mortality should therefore be considerably higher. For instance, in 1898 three deaths came within this category, which just doubles the mortality; and in 1899 the additional deaths were six, bringing the mortality up to two-and-a-half times that indicated in the table. When, however, the additional deaths are included the results remain sufficiently striking, if two assumptions are granted--(1) that all the persons treated have been bitten by rabid animals; (2) that a large proportion of persons so bitten usually have hydrophobia. Unfortunately, both these assumptions lack proof, and therefore the evidence of the efficacy of the treatment cannot be said to satisfy a strictly scientific standard. With regard to the first point, the patients are divided into three categories--(1) those bitten by an animal the rabidity of which is proved by the development of rabies in other animals bitten by it or inoculated from its spinal cord; (2) those bitten by an animal pronounced rabid on a veterinary examination; (3) those bitten by an animal suspected of being rabid. The number of patients in each category in 1898 was (1) 141, (2) 855, (3) 469; and in 1899 it was (1) 152, (2) 1099, (3) 363. As might be expected, the vast majority came under the second and third heads, in which the evidence of rabidity is doubtful or altogether lacking. With regard to the second point, the proportion of persons bitten by rabid animals who ordinarily develop hydrophobia has only been "estimated" from very inadequate data. Otto Bollinger from a series of collected statistics states that before the introduction of the Pasteur treatment, of patients bitten by dogs undoubtedly rabid 47% died, the rate being 33% in those whose wounds had been cauterized and 83% when there had been no local treatment. If the number of rabid dogs be compared with the deaths from hydrophobia in any year or series of years, it can hardly be very high. For instance, in 1895, 668 dogs, besides other animals, were killed and certified to be rabid in England, and the deaths from hydrophobia were twenty. Of course this proves nothing, as the number of persons bitten is not known, but the difference between the amount of rabies and of hydrophobia is suggestively great in view of the marked propensity of rabid dogs to bite, nor is it accounted for by the fact that some of the persons bitten were treated at the Institut Pasteur. A comparison of the annual mortality from hydrophobia in France before and after the introduction of the treatment would afford decisive evidence as to its efficacy; but unfortunately no such comparison can be made for lack of vital statistics in that country. The experience of the Paris hospitals, however, points to a decided diminution of mortality. On the whole it must be said, in the absence of further data, that the Pasteur treatment certainly diminishes the danger of hydrophobia from the bites of rabid animals.
More recently treatment with an anti-rabic serum has been suggested (see PARASITIC DISEASES). Victor Babes and Lepp and later Guido Tizzoni and Eugenio Centanni have worked out a method of serum treatment curative and protective. In this method not the rabic poison itself, as in the Pasteur treatment, but the protective substance formed is injected into the tissues. The serum of a vaccinated animal is capable of neutralizing the power of the virus of rabies not only when mixed with the virus before injection but even when injected simultaneously or within twenty-four hours after the introduction of the virus. These authors showed that the serum of a rabbit protects a rabbit better than does the serum of a dog, and vice versa. At the end of twenty days' injections they found they could obtain such a large quantity of anti-rabic substance in the serum of an animal, that even 1 part of serum to 25,000 of the body weight would protect an animal. This process differs from that of Pasteur in so far as that in place of promoting the formation of the antidote within the body of the patient, by a process of vaccination with progressively stronger and stronger virus, this part of the process is carried on in an animal, Babes using the dog and Centanni the sheep, the blood serum of which is injected. This method of vaccination is useful as a protective to those in charge of kennels.
Muzzling order in England.
The attempt to stamp out rabies in Great Britain was an experiment undertaken by the government in the public interest. The principal means adopted were the muzzling of dogs in infected areas, and prolonged quarantine for imported animals. The efficacy of dog-muzzling in checking the spread of rabies and diminishing its prevalence has been repeatedly proved in various countries. Liable as other animals may be to the disease, in England at least the dog is pre-eminently the vehicle of contagion and the great source of danger to human beings. There is a difference of opinion on the way in which muzzling acts, though there can be none as to the effect it produces in reducing rabies. Probably it acts rather by securing the destruction of ownerless and stray--which generally includes rabid--dogs than by preventing biting; for though it may prevent snapping, even the wire-cage muzzle does not prevent furious dogs from biting, and it is healthy, not rabid, dogs that wear the muzzle. It has therefore been suggested that a collar would have the same effect, if all collarless dogs were seized; but the evidence goes to show that it has not, perhaps because rabid dogs are more likely to stray from home with their collars, which are constantly worn, than with muzzles which are not, and so escape seizure. Moreover, it is much easier for the police to see whether a dog is wearing a muzzle or not than it is to make sure about the collar. However this may be, the muzzle has proved more efficacious, but it was not applied systematically in England until a late date. Sometimes the regulations were in the hands of the government, and sometimes they were left to local authorities; in either case they were allowed to lapse as soon as rabies had died down. In April 1897 the Board of Agriculture entered on a systematic attempt to exterminate rabies by the means indicated. The plan was to enforce muzzling over large areas in which the disease existed, and to maintain it for six months after the occurrence of the last case. In spite of much opposition and criticism, this was resolutely carried out under Mr Walter Long, the responsible minister, and met with great success. By the spring of 1899--that is, in two years--the disease had disappeared in Great Britain, except for one area in Wales; and, with this exception, muzzling was everywhere relaxed in October 1899. It was taken off in Wales also in the following May, no case having occurred since November 1899. Rabies was then pronounced extinct. During the summer of 1900, however, it reappeared in Wales, and several counties were again placed under the order. The year 1901 was the third in succession in which no death from hydrophobia was registered in the United Kingdom. In the ten years preceding 1899, 104 deaths were registered, the death-rate reaching 30 in 1889 and averaging 29 annually. In 1902 two deaths from hydrophobia were registered. From that date to June 1909 (the latest available for the purpose of this article) no death from hydrophobia was notified in the United Kingdom.
See _Annales de l'Institut Pasteur_, from 1886; _Journal of the Board of Agriculture_, 1899; Makins, "Hydrophobia," in Treves's _System of Surgery_; Woodhead, "Rabies," in Allbutt's _System of Medicine_.
HYDROSPHERE (Gr. [Greek: hydôr], water, and [Greek: sphaira], sphere), in physical geography, a name given to the whole mass of the water of the oceans, which fills the depressions in the earth's crust, and covers nearly three-quarters of its surface. The name is used in distinction from the atmosphere, the earth's envelope of air, the lithosphere (Gr. [Greek: lithos], rock) or solid crust of the earth, and the centrosphere or interior mass within the crust. To these "spheres" some writers add, by figurative usage, the terms "biosphere," or life-sphere, to cover all living things, both animals and plants, and "psychosphere," or mind-sphere, covering all the products of human intelligence.
HYDROSTATICS (Gr. [Greek: hydôr], water, and the root [Greek: sta]-, to cause to stand), the branch of hydromechanics which discusses the equilibrium of fluids (see HYDROMECHANICS).
HYDROXYLAMINE, NH2OH, or hydroxy-ammonia, a compound prepared in 1865 by W. C. Lossen by the reduction of ethyl nitrate with tin and hydrochloric acid. In 1870 E. Ludwig and T. H. Hein (_Chem. Centralblatt_, 1870, 1, p. 340) obtained it by passing nitric oxide through a series of bottles containing tin and hydrochloric acid, to which a small quantity of platinum tetrachloride has been added; the acid liquid is poured off when the operation is completed, and sulphuretted hydrogen is passed in; the tin sulphide is filtered off and the filtrate evaporated. The residue is extracted by absolute alcohol, which dissolves the hydroxylamine hydrochloride and a little ammonium chloride; this last substance is removed as ammonium platino-chloride, and the residual hydroxylamine hydrochloride is recrystallized. E. Divers obtains it by mixing cold saturated solutions containing one molecular proportion of sodium nitrate, and two molecular proportions of acid sodium sulphite, and then adding a saturated solution of potassium chloride to the mixture. After standing for twenty-four hours, hydroxylamine potassium disulphonate crystallizes out. This is boiled for some hours with water and the solution cooled, when potassium sulphate separates first, and then hydroxylamine sulphate. E. Tafel (_Zeit. anorg. Chem._, 1902, 31, p. 289) patented an electrolytic process, wherein 50% sulphuric acid is treated in a divided cell provided with a cathode of amalgamated lead, 50% nitric acid being gradually run into the cathode compartment. Pure anhydrous hydroxylamine has been obtained by C. A. Lobry de Bruyn from the hydrochloride, by dissolving it in absolute methyl alcohol and then adding sodium methylate. The precipitated sodium chloride is filtered, and the solution of hydroxylamine distilled in order to remove methyl alcohol, and finally fractionated under reduced pressure. The free base is a colourless, odourless, crystalline solid, melting at about 30° C., and boiling at 58° C. (under a pressure of 22 mm.). It deliquesces and oxidizes on exposure, inflames in dry chlorine and is reduced to ammonia by zinc dust. Its aqueous solution is strongly alkaline, and with acids it forms well-defined stable salts. E. Ebler and E. Schott (_J. pr. Chem._, 1908, 78, p. 289) regard it as acting with the formula NH2·OH towards bases, and as NH3:O towards acids, the salts in the latter case being of the oxonium type. It is a strong reducing agent, giving a precipitate of cuprous oxide from alkaline copper solutions at ordinary temperature, converting mercuric chloride to mercurous chloride, and precipitating metallic silver from solutions of silver salts. With aldehydes and ketones it forms oximes (q.v.). W. R. Dunstan (_Jour. Chem. Soc._, 1899, 75, p. 792) found that the addition of methyl iodide to a methyl alcohol solution of hydroxylamine resulted in the formation of trimethyloxamine, N(CH3)3O.
Many substituted hydroxylamines are known, substitution taking place either in the [alpha] or [beta] position
[beta] [alpha] (NH2 · OH^).
[beta]-phenylhydroxyl-amine, C6H5NH·OH·, is obtained in the reduction of nitrobenzene in neutral solution (e.g. by the action of the aluminium-mercury couple and water), but better, according to C. Goldschmidt (_Ber._, 1896, 29, p. 2307) by dissolving nitrobenzene in ten times its weight of ether containing a few cubic centimetres of water, and heating with excess of zinc dust and anhydrous calcium chloride for three hours on a water bath. It also appears as an intermediate product in the electrolytic reduction of nitrobenzene in sulphuric acid solution. By gentle oxidation it yields nitrosobenzene. Derivatives of the type R2N·OH result in the action of the Grignard reagent on amyl nitrite. Dihydroxy-ammonia or nitroxyl, NH(OH)2, a very unstable and highly reactive substance, has been especially studied by A. Angeli (see A. W. Stewart, _Recent Advances in Physical and Inorganic Chemistry_, 1909).
HYDROZOA, one of the most widely spread and prolific groups of aquatic animals. They are for the most part marine in habitat, but a familiar fresh-water form is the common _Hydra_ of ponds and ditches, which gives origin to the name of the class. The Hydrozoa comprise the hydroids, so abundant on all shores, most of which resemble vegetable organisms to the unassisted eye; the hydrocorallines, which, as their name implies, have a massive stony skeleton and resemble corals; the jelly-fishes so called; and the Siphonophora, of which the species best known by repute is the so-called "Portuguese man-of-war" (_Physalia_), dreaded by sailors on account of its terrible stinging powers.
In external form and appearance the Hydrozoa exhibit such striking differences that there would seem at first sight to be little in common between the more divergent members of the group. Nevertheless there is no other class in the animal kingdom with better marked characteristics, or with more uniform morphological peculiarities underlying the utmost diversity of superficial characters.
All Hydrozoa, in the first place, exhibit the three structural features distinctive of the Coelentera (q.v.). (1) The body is built up of two layers only, an external protective and sensory layer, the ectoderm, and an internal digestive layer, the endoderm. (2) The body contains but a single internal cavity, the coelenteron or gastrovascular space, which may be greatly ramified, but is not shut off into cavities distinct from the central digestive space. (3) The generative cells are produced in either the ectoderm or endoderm, and not in a third layer arising in the embryo, distinct from the two primary layers; in other words, there is no mesoderm or coelom.
To these three characters the Hydrozoa add a fourth which is distinctive of the subdivision of the Coelenterata termed the Cnidaria; that is to say, they always possess peculiar stinging organs known as nettle-cells, or _nematocysts_ (_Cnidae_), each produced in a cell forming an integral part of the animal's tissues. The Hydrozoa are thus shown to belong to the group of Coelenterata Cnidaria, and it remains to consider more fully their distinctive features, and in particular those which mark them off from the other main division of the Cnidaria, the Anthozoa (q.v.), comprising the corals and sea-anemones.
The great diversity, to which reference has already been made, in the form and structure of the Hydrozoa is due to two principal causes. In the first place, we find in this group two distinct types of person or individual, the polyp and the medusa (qq.v.), each capable of a wide range of variations; and when both polyp and medusa occur in the life-cycle of the same species, as is frequently the case, the result is an alternation of generations of a type peculiarly characteristic of the class. In the second place, the power of non-sexual reproduction by budding is practically of universal occurrence among the Hydrozoa, and by the buds failing to separate from the parent stock, colonies are produced, more or less complicated in structure and often of great size. We find that polyps may either bud other polyps or may produce medusae, and that medusae may bud medusae, though never, apparently, polyps. Hence we have a primary subdivision of the colonies of Hydrozoa into those produced by budding of polyps and those produced by budding of medusae. The former may contain polyp-persons and medusa-persons, either one kind alone or both kinds combined; the latter will contain only medusa-persons variously modified.
The morphology of the Hydrozoa reduces itself, therefore, to a consideration of the morphology of the polyp, of the medusa and of the colony. Putting aside the last-named, for a detailed account of which see HYDROMEDUSAE, we can best deal with the peculiarities of the polyp and medusa from a developmental point of view.
In the development of the Hydrozoa, and indeed of the Cnidaria generally, the egg usually gives rise to an oval larva which swims about by means of a coating of cilia on the surface of the body. This very characteristic larva is termed a _planula_, but though very uniform externally, the planulae of different species, or of the same species at different periods, do not always represent the same stage of embryonic development internally. On examining more minutely the course of the development, it is found that the ovum goes through the usual process of cleavage, always total and regular in this group, and so gives rise to a hollow sphere or ovoid with the wall composed of a single layer of cells, and containing a spacious cavity, the blastocoele or segmentation-cavity. This is the _blastula_ stage occurring universally in all Metazoa, probably representing an ancestral Protozoan colony in phylogeny. Next the blastula gives rise to an internal mass of cells (fig. 1, hy) which come from the wall either by immigration (fig. 1, A) or by splitting off (delamination). The formation of an inner cell-mass converts the single-layered blastula (monoblastula) into a double-layered embryo (diblastula) which may be termed a parenchymula, since at first the inner cell-mass forms an irregular parenchyma which may entirely fill up and obliterate the segmentation cavity (fig. 1, B). At a later stage, however, the cells of the inner mass arrange themselves in a definite layer surrounding an internal cavity (fig. 1, C, al), which soon acquires an opening to the exterior at one pole, and so forms the characteristic embryonic stage of all Enterozoa known as the _gastrula_ (fig. 2). In this stage the body is composed of two layers, ectoderm (d) externally, and endoderm (c) internally, surrounding a central cavity, the _archenteron_ (b), which communicates with the exterior by a pore (a), the _blastopore_.
Thus a planula larva may be a blastula, or but slightly advanced beyond this stage, or it may be (and most usually is) a parenchymula; or in some cases (Scyphomedusae) it may be a gastrula. It should be added that the process of development, the gastrulation as it is termed, may be shortened by the immigration of cells taking place at one pole only, and in a connected layer with orderly arrangement, so that the gastrula stage is reached at once from the blastula without any intervening parenchymula stage. This is a process of gastrulation by invagination which is found in all animals above the Coelenterata, but which is very rare in the Cnidaria, and is known only in the Scyphomedusae amongst the Hydrozoa.
After the gastrula stage, which is found as a developmental stage in all Enterozoa, the embryo of the Hydrozoa proceeds to develop characters which are peculiar to the Coelenterata only. Round the blastopore hollow outgrowths, variable in number, arise by the evagination of the entire body-wall, both ectoderm and endoderm. Each outgrowth contains a prolongation of the archenteric cavity (compare figs. 2 and 3, A). In this way is formed a ring of tentacles, the most characteristic organs of the Cnidaria. They surround a region which is termed the peristome, and which contains in the centre the blastopore, which becomes the adult mouth. The archenteron becomes the gastrovascular system or coelenteron. Between the ectoderm and endoderm a gelatinous supporting layer, termed the mesogloea, makes its appearance. The gastrula has now become an _actinula_, which may be termed the distinctive larva of the Cnidaria, and doubtless represents in a transitory manner the common ancestor of the group. In no case known, however, does the actinula become the adult, sexually mature individual, but always undergoes further modifications, whereby it develops into either a polyp or a medusa.
To become a polyp, the actinula (fig. 3, A) becomes attached to some firm object by the pole farthest from the mouth, and its growth preponderates in the direction of the principal axis, that is to say, the axis passing through the mouth (fig. 3, _a-b_). As a result the body becomes columnar in form (fig. 3, B), and without further change passes into the characteristic polyp-form (see POLYP).
It is convenient to distinguish two types of polyp by the names hydro polyp and anthopolyp, characteristic of the Hydrozoa and Anthozoa respectively. In the hydropolyp the body is typically elongated, the height of the column being far greater than the diameter. The peristome is relatively small and the mouth is generally raised on a projecting spout or _hypostome_. The ectoderm loses entirely the ciliation which it had in the planula and actinula stages and commonly secretes on its external surface a protective or supporting investment, the perisarc. Contrasting with this, the anthopolyp is generally of squat form, the diameter often exceeding the height; the peristome is wide, a hypostome is lacking, and the ectoderm, or so much of it as is exposed, i.e. not covered by secretion of skeletal or other investment, retains its ciliation throughout life. The internal structural differences are even more characteristic. In the hydropolyp the blastopore of the embryo forms the adult mouth situated at the extremity of the hypostome, and the ectoderm and endoderm meet at this point. In the anthopolyp the blastopore is carried inwards by an in-pushing of the body-wall of the region of the peristome, so that the adult mouth is an opening leading into a short ectodermal oesophagus or stomodaeum, at the bottom of which is the blastopore. Further, in the hydropolyp the digestive cavity either remains simple and undivided and circular in transverse section, or may show ridges projecting internally, which in this case are formed of endoderm alone, without any participation of the mesogloea. In the anthopolyp, on the other hand, the digestive cavity is always subdivided by so-called mesenteries, in-growths of the endoderm containing vertical lamellae of mesogloea (see ANTHOZOA). In short, the hydropolyp is characterized by a more simple type of organization than the anthopolyp, and is in most respects less modified from the actinula type of structure.
Returning now to the actinula, this form may, as already stated, develop into a medusa, a type of individual found only in the Hydrozoa, as here understood. To become a medusa, the actinula grows scarcely at all in the direction of the principal axis, but greatly along a plane at right angles to it. Thus the body becomes umbrella-shaped, the concave side representing the peristome, and the convex side the column, of the polyp. Hence the tentacles are found at the edge of the umbrella, and the hypostome forms usually a projecting tube, with the mouth at the extremity, forming the _manubrium_ or handle of the umbrella. The medusa has a pronounced radial symmetry, and the positions of the primary tentacles, usually four in number, mark out the so-called _radii_, alternating with which are four _interradii_. The ectoderm retains its ciliation only in the sensory organs. The mesogloea becomes enormously increased in quantity (hence the popular name "jelly-fish"), and in correlation with this the endoderm-layer lining the coelenteron becomes pressed together in the interradial areas and undergoes concrescence, forming a more or less complicated gastrovascular system (see MEDUSA). It is sufficient to state here that the medusa is usually a free-swimming animal, floating mouth downwards on the open seas, but in some cases it may be attached by its aboral pole, like a polyp, to some firm basis, either temporarily or permanently.
Thus the development of the two types of individual seen in the Hydrozoa may be summarized as follows:--
/ Egg | | Free | Blastula | | "Planula" < Parenchymula | | Stage | Gastrula | | \ Actinula / \ / \ Polyp Medusa
This development, though probably representing the primitive sequence of events, is never actually found in its full extent, but is always abbreviated by omission or elimination of one or more of the stages. We have already seen that the parenchymula stage is passed over when the gastrulation is of the invaginate type. On the other hand, the parenchymula may develop directly into the actinula or even into the polyp, with suppression of the intervening steps. Great apparent differences may also be brought about by variations in the period at which the embryo is set free as a larva, and since two free-swimming stages, planula and actinula, are unnecessary, one or other of them is always suppressed. A good example of this is seen in two common genera of British hydroids, _Cordylophora_ and _Tabularia_. In _Cordylophora_ the embryo is set free at the parenchymula stage as a planula which fixes itself and develops into a polyp, both gastrula and actinula stages being suppressed. In _Tubularia_, on the other hand, the parenchymula develops into an actinula within the maternal tissues, and is then set free, creeps about for a time, and after fixing itself, changes into a polyp; hence in this case the planula-stage, as a free larva, is entirely suppressed.
The Hydrozoa may be defined, therefore, as Cnidaria in which two types of individual, the polyp and the medusa, may be present, each type developed along divergent lines from the primitive actinula form. The polyp (hydropolyp) is of simple structure and never has an ectodernal oesophagus or mesenteries.[1] The general ectoderm loses its cilia, which persist only in the sensory cells, and it frequently secretes external protective or supporting structures. An internal mesogloeal skeleton is not found.
The class is divisible into two main divisions or sub-classes, Hydromedusae and Scyphomedusae, of which definitions and detailed systematic accounts will be found under these headings.
GENERAL WORKS ON HYDROZOA.--C. Chun, "Coelenterata (Hohlthiere)," _Bronn's Klassen und Ordnungen des Thier-Reichs_ ii. 2 (1889 et seq.); Y. Delage, and E. Hérouard, _Traité de zoologie concrète_, ii. part 2, _Les Coelentérés_ (1901); G. H. Fowler, "The Hydromedusae and Scyphomedusae" in E. R. Lankester's _Treatise on Zoology_, ii. chapters iv. and v. (1900); S. J. Hickson, "Coelenterata and Ctenophora," _Cambridge Natural History_, i. chapters x.-xv. (1906). (E. A. M.)
FOOTNOTE:
[1] See further under SCYPHOMEDUSAE.
HYENA, a name applicable to all the representatives of the mammalian family _Hyaenidae_, a group of Carnivora (q.v.) allied to the civets. From all other large Carnivora except the African hunting-dog, hyenas are distinguished by having only four toes on each foot, and are further characterized by the length of the fore-legs as compared with the hind pair, the non-retractile claws, and the enormous strength of the jaws and teeth, which enables them to break the hardest bones and to retain what they have seized with unrelaxing grip.
The striped hyena (_Hyaena striata_) is the most widely distributed species, being found throughout India, Persia, Asia Minor, and North and East Africa, the East African form constituting a distinct race, _H. striata schillingsi_; while there are also several distinct Asiatic races. The species resembles a wolf in size, and is greyish-brown In colour, marked with indistinct longitudinal stripes of a darker hue, while the legs are transversely striped. The hairs on the body are long, especially on the ridge of the neck and back, where they form a distinct mane, which is continued along the tail. Nocturnal in habits, it prefers by day the gloom of caves and ruins, or of the burrows which it occasionally forms, and issues forth at sunset, when it commences its unearthly howling. When the animal is excited, the howl changes into what has been compared to demoniac laughter, whence the name of "laughing-hyena." These creatures feed chiefly on carrion, and thus perform useful service by devouring remains which might otherwise pollute the air. Even human dead are not safe from their attacks, their powerful claws enabling them to gain access to newly interred bodies in cemeteries. Occasionally (writes Dr W. T. Blanford) sheep or goats, and more often dogs, are carried off, and the latter, at all events, are often taken alive to the animal's den. This species appears to be solitary in habits, and it is rare to meet with more than two together. The cowardice of this hyena is proverbial; despite its powerful teeth, it rarely attempts to defend itself. A very different animal is the spotted hyena, _Hyaena (Crocuta) crocuta_, which has the sectorial teeth of a more cat-like type, and is marked by dark-brown spots on a yellowish ground, while the mane is much less distinct. At the Cape it was formerly common, and occasionally committed great havoc among the cattle, while it did not hesitate to enter the Kaffir dwellings at night and carry off children sleeping by their mothers. By persistent trapping and shooting, its numbers have now been considerably reduced, with the result, however, of making it exceedingly wary, so that it is not readily caught in any trap with which it has had an opportunity of becoming acquainted. Its range extends from Abyssinia to the Cape. The Abyssinian form has been regarded as a distinct species, under the name of _H. liontiewi_, but this, like various more southern forms, is but regarded as a local race. The brown hyena (_H. brunnea_) is South African, ranging to Angola on the west and Kilimanjaro on the east. In size it resembles the striped hyena, but differs in appearance, owing to the fringe of long hair covering the neck and fore part of the back. The general hue is ashy-brown, with the hair lighter on the neck (forming a collar), chest and belly; while the legs are banded with dark brown. This species is not often seen, as it remains concealed during the day. Those frequenting the coast feed on dead fish, crabs and an occasional stranded whale, though they are also a danger to the sheep and cattle kraal. Strand-wolf is the local name at the Cape.
Although hyenas are now confined to the warmer regions of the Old World, fossil remains show that they had a more northerly range during Tertiary times; the European cave-hyena being a form of the spotted species, known as _H. crocuta spelaea_. Fossil hyenas occur in the Lower Pliocene of Greece, China, India, &c.; while remains indistinguishable from those of the striped species have been found in the Upper Pliocene of England and Italy.
HYÈRES, a town in the department of the Var in S.E. France, 11 m. by rail E. of Toulon. In 1906 the population of the commune was 17,790, of the town 10,464; the population of the former was more than doubled in the last decade of the 19th century. Hyères is celebrated (as is also its fashionable suburb, Costebelle, nearer the seashore) as a winter health resort. The town proper is situated about 2½ m. from the seashore, and on the south-western slope of a steep hill (669 ft., belonging to the Maurettes chain, 961 ft.), which is one of the westernmost spurs of the thickly wooded Montagnes des Maures. It is sheltered from the north-east and east winds, but is exposed to the cold north-west wind or _mistral_. Towards the south and south-east a fertile plain, once famous for its orange groves, but now mainly covered by vineyards and farms, stretches to the sea, while to the south-west, across a narrow valley, rises a cluster of low hills, on which is the suburb of Costebelle. The older portion of the town is still surrounded, on the north and east, by its ancient, though dilapidated medieval walls, and is a labyrinth of steep and dirty streets. The more modern quarter which has grown up at the southern foot of the hill has handsome broad boulevards and villas, many of them with beautiful gardens, filled with semi-tropical plants. Among the objects of interest in the old town are: the house (Rue Rabaton, 7) where J. B. Massillon (1663-1742), the famous pulpit orator, was born; the parish church of St Louis, built originally in the 13th century by the Cordelier or Franciscan friars, but completely restored in the earlier part of the 19th century; and the site of the old château, on the summit of the hill, now occupied by a villa. The plain between the new town and the sea is occupied by large nurseries, an excellent _jardin d'acclimatation_, and many market gardens, which supply Paris and London with early fruits and vegetables, especially artichokes, as well as with roses in winter. There are extensive salt beds (_salines_) both on the peninsula of Giens, S. of the town, and also E. of the town. To the east of the Giens peninsula is the fine natural harbour of Hyères, as well as three thinly populated islands (the Stoechades of the ancients), Porquerolles, Port Cros and Le Levant, which are grouped together under the common name of Îles d'Hyères.
The town of Hyères seems to have been founded in the 10th century, as a place of defence against pirates, and takes its name from the aires (_hierbo_ in the Provençal dialect), or threshing-floors for corn, which then occupied its site. It passed from the possession of the viscounts of Marseilles to Charles of Anjou, count of Provence, and brother of St Louis (the latter landed here in 1254, on his return from Egypt). The château was dismantled by Henri IV., but thanks to its walls, the town resisted in 1707 an attack made by the duke of Savoy.
See Ch. Lenthéric, _La Provence Maritime ancienne et moderne_ (chap. 5) (Paris, 1880). (W. A. B. C.)
HYGIEIA, in Greek mythology, the goddess of health. It seems probable that she was originally an abstraction, subsequently personified, rather than an independent divinity of very ancient date. The question of the original home of her worship has been much discussed. The oldest traces of it, so far as is known at present, are to be found at Titane in the territory of Sicyon, where she was worshipped together with Asclepius, to whom she appears completely assimilated, not an independent personality. Her cult was not introduced at Epidaurus till a late date, and therefore, when in 420 B.C. the worship of Asclepius was introduced at Athens coupled with that of Hygieia, it is not to be inferred that she accompanied him from Epidaurus, or that she is a Peloponnesian importation at all. It is most probable that she was invented at the time of the introduction of Asclepius, after the sufferings caused by the plague had directed special attention to sanitary matters. The already existing worship of Athena Hygieia had nothing to do with Hygieia the goddess of health, but merely denoted the recognition of the power of healing as one of the attributes of Athena, which gradually became crystallized into a concrete personality. At first no special relationship existed between Asclepius and Hygieia, but gradually she came to be regarded as his daughter, the place of his wife being already secured by Epione. Later Orphic hymns, however, and Herodas iv. 1-9, make her the wife of Asclepius. The cult of Hygieia then spread concurrently with that of Asclepius, and was introduced at Rome from Epidaurus in 293, by which time she may have been admitted (which was not the case before) into the Epidaurian family of the god. Her proper name as a Romanized Greek importation was Valetudo, but she was gradually identified with Salus, an older genuine Italian divinity, to whom a temple had already been erected in 302. While in classical times Asclepius and Hygieia are simply the god and goddess of health, in the declining years of paganism they are protecting divinities generally, who preserve mankind not only from sickness but from all dangers on land and sea. In works of art Hygieia is represented, together with Asclepius, as a maiden of benevolent appearance, wearing the chiton and giving food or drink to a serpent out of a dish.
See the article by H. Lechat in Daremberg and Saglio's _Dictionnaire des antiquités_, with full references to authorities; and E. Thrämer in Roscher's _Lexikon der Mythologie_, with a special section on the modern theories of Hygieia.
HYGIENE (Fr. _hygiène_, from Gr. [Greek: hygiainein], to be healthy), the science of preserving health, its practical aim being to render "growth more perfect, decay less rapid, life more vigorous, death more remote." The subject is thus a very wide one, embracing all the agencies which affect the physical and mental well-being of man, and it requires acquaintance with such diverse sciences as physics, chemistry, geology, engineering, architecture, meteorology, epidemiology, bacteriology and statistics. On the personal or individual side it involves consideration of the character and quality of food and of water and other beverages; of clothing; of work, exercise and sleep; of personal cleanliness, of special habits, such as the use of tobacco, narcotics, &c.; and of control of sexual and other passions. In its more general and public aspects it must take cognizance of meteorological conditions, roughly included under the term climate; of the site or soil on which dwellings are placed; of the character, materials and arrangement of dwellings, whether regarded individually or in relation to other houses among which they stand; of their heating and ventilation; of the removal of excreta and other effete matters; of medical knowledge relating to the incidence and prevention of disease; and of the disposal of the dead.
These topics will be found treated in such articles as DIETETICS, FOOD, FOOD-PRESERVATION, ADULTERATION, WATER, HEATING, VENTILATION, SEWERAGE, BACTERIOLOGY, HOUSING, CREMATION, &c. For legal enactments which concern the sanitary well-being of the community, see PUBLIC HEALTH.
HYGINUS, eighth pope. It was during his pontificate (c. 137-140) that the gnostic heresies began to manifest themselves at Rome.
HYGINUS (surnamed GROMATICUS, from _gruma_, a surveyor's measuring-rod), Latin writer on land-surveying, flourished in the reign of Trajan (A.D. 98-117). Fragments of a work on legal boundaries attributed to him will be found in C. F. Lachmann, _Gromatici Veteres_, i. (1848).
A treatise on Castrametation (_De Munitionibus Castrorum_), also attributed to him, is probably of later date, about the 3rd century A.D. (ed. W. Gemoll, 1879; A. von Domaszewski, 1887).
HYGINUS, GAIUS JULIUS, Latin author, a native of Spain (or Alexandria), was a pupil of the famous Cornelius Alexander Polyhistor and a freedman of Augustus, by whom he was made superintendent of the Palatine library (Suetonius, _De Grammaticis_, 20). He is said to have fallen into great poverty in his old age, and to have been supported by the historian Clodius Licinus. He was a voluminous author, and his works included topographical and biographical treatises, commentaries on Helvius Cinna and the poems of Virgil, and disquisitions on agriculture and bee-keeping. All these are lost.
Under the name of Hyginus two school treatises on mythology are extant: (1) _Fabularum Liber_, some 300 mythological legends and celestial genealogies, valuable for the use made by the author of the works of Greek tragedians now lost; (2) _De Astronomia_, usually called _Poetica Astronomica_, containing an elementary treatise on astronomy and the myths connected with the stars, chiefly based on the [Greek: Katasterismoi] of Eratosthenes. Both are abridgments and both are by the same hand; but the style and Latinity and the elementary mistakes (especially in the rendering of the Greek originals) are held to prove that they cannot have been the work of so distinguished a scholar as C. Julius Hyginus. It is suggested that these treatises are an abridgment (made in the latter half of the 2nd century) of the _Genealogiae_ of Hyginus by an unknown grammarian, who added a complete treatise on mythology.
EDITIONS.--_Fabulae_, by M. Schmidt (1872); _De Astronomia_, by B. Bunte (1875); see also Bunte, _De C. Julii Hygini, Augusti Liberti, Vita et Scriptis_ (1846).
HYGROMETER (Gr. [Greek: hygros], moist, [Greek: metron], a measure), an instrument for measuring the absolute or relative amount of moisture in the atmosphere; an instrument which only qualitatively determines changes in the humidity is termed a "hygroscope." The earlier instruments generally depended for their action on the contraction or extension of substances when exposed to varying degrees of moisture; catgut, hair, twisted cords and wooden laths, all of which contract with an increase in the humidity and vice versa, being the most favoured materials. The familiar "weather house" exemplifies this property. This toy consists of a house provided with two doors, through which either a man or woman appears according as the weather is about to be wet or fine. This action is effected by fixing a catgut thread to the base on which the figures are mounted, in such a manner that contraction of the thread rotates the figures so that the man appears and extension so that the woman appears.
Many of the early forms are described in C. Hutton, _Math. and Phil. Dictionary_ (1815). The modern instruments, which utilize other principles, are described in METEOROLOGY: II. _Methods and Apparatus_.
HYKSOS, or "SHEPHERD KINGS," the name of the earliest invaders of Egypt of whom we have definite evidence in tradition. Josephus (c. _Apion._ i. 14), who identifies the Hyksos with the Israelites, preserves a passage from the second book of Manetho giving an account of them. (It may be that Josephus had it, not direct from Manetho's writings, but through the garbled version of some Alexandrine compiler.) In outline it is as follows. In the days of a king of Egypt named Timaeus the land was suddenly invaded from the east by men of ignoble race, who conquered it without a struggle, destroyed cities and temples, and slew or enslaved the inhabitants. At length they elected a king named Salatis, who, residing at Memphis, made all Egypt tributary, and established garrisons in different parts, especially eastwards, fearing the Assyrians. He built also a great fortress at Avaris, in the Sethroite nome, east of the Bubastite branch of the Nile. Salatis was followed in succession by Beon, Apachnas, Apophis, Jannas and Asses. These six kings reigned 198 years and 10 months, and all aimed at extirpating the Egyptians. Their whole race was named Hyksos, i.e. "shepherd kings," and some say they were Arabs (another explanation found by Josephus is "captive shepherds"). When they and their successors had held Egypt for 511 years, the kings of the Thebais and other parts of Egypt rebelled, and a long and mighty war began. Misphragmuthosis worsted the "Shepherds" and shut them up in Avaris; and his son Thutmosis, failing to capture the stronghold, allowed them to depart; whereupon they went forth, 240,000 in number, established themselves in Judea and built Jerusalem.
In Manetho's list of kings, the six above named (with many variations in detail) form the XVth dynasty, and are called "six foreign Phoenician kings." The XVIth dynasty is of thirty-two "Hellenic (_sic?_) shepherd kings," the seventeenth is of "shepherds and Theban kings" (reigning simultaneously). The lists vary greatly in different versions, but the above seems the most reasonable selection of readings to be made. For "Hellenic" see below. The supposed connexion with the Israelites has made the problem of the Hyksos attractive, but light is coming upon it very slowly. In 1847 E. de Rougé proved from a fragment of a story in the papyri of the British Museum, that Apopi was one of the latest of the Hyksos kings, corresponding to Aphobis; he was king of the "pest" and suppressed the worship of the Egyptian gods, and endeavoured to make the Egyptians worship his god Setekh or Seti; at the same time an Egyptian named Seqenenre reigned in Thebes, more or less subject to Aphobis. The city of Hawari (Avaris) was also mentioned in the fragment.
In 1850 a record of the capture of this city from the Hyksos by Ahmosi, the founder of the eighteenth dynasty, was discovered by the same scholar. A large class of monuments was afterwards attributed to the Hyksos, probably in error. Some statues and sphinxes, found in 1861 by Mariette at Tanis (in the north-east of the Delta), which had been usurped by later kings, had peculiar "un-Egyptian" features. One of these bore the name of Apopi engraved lightly on the shoulder; this was evidently a usurper's mark, but from the whole circumstances it was concluded that these, and others of the same type of features found elsewhere, must have belonged to the Hyksos. This view held the field until 1893, when Golénischeff produced an inferior example bearing its original name, which showed that in this case it represented Amenemhe III. In consequence it is now generally believed that they all belong to the twelfth dynasty. Meanwhile a headless statue of a king named Khyan, found at Bubastis, was attributed on various grounds to the Hyksos, the soundest arguments being his foreign name and the boastful un-Egyptian epithet "beloved of his _ka_," where "beloved of Ptah" or some other god was to be expected. His name was immediately afterwards recognized on a lion found as far away from Egypt as Bagdad. Flinders Petrie then pointed out a group of kings named on scarabs of peculiar type, which, including Khyan, he attributed to the period between the Old Kingdom and the New, while others were in favour of assigning them all to the Hyksos, whose appellation seemed to be recognizable in the title Hek-khos, "ruler of the barbarians," borne by Khyan. The extraordinary importance of Khyan was further shown by the discovery of his name on a jar-lid at Cnossus in Crete. Semitic features were pointed out in the supposed Hyksos names, and Petrie was convinced of their date by his excavations of 1905-1906 in the eastern Delta. Avaris is generally assigned to the region towards Pelusium on the strength of its being located in the Sethroite nome by Josephus, but Petrie thinks it was at Tell el-Yahudiyeh (Yehudia), where Hyksos scarabs are common. From the remains of fortifications there he argues that the Hyksos were uncivilized desert people, skilled in the use of the bow, and must thus have destroyed by their archery the Egyptian armies trained to fight hand-to-hand; further, that their hordes were centered in Syria, but were driven thence by a superior force in the East to take refuge in the islands and became a sea-power--whence the strange description "Hellenic" in Manetho, which most editors have corrected to [Greek: alloi], "others." Besides the statue of Khyan, blocks of granite with the name of Apopi have been found in Upper Egypt at Gebelen and in Lower Egypt at Bubastis. The celebrated Rhind mathematical papyrus was copied in the reign of an Apopi from an original of the time of Amenemhe III. Large numbers of Hyksos scarabs are found in Upper and Lower Egypt, and they are not unknown in Palestine. Khyan's monuments, inconspicuous as they are, actually extend over a wider area--from Bagdad to Cnossus--than those of any other Egyptian king.
It is certain that this mysterious people were Asiatic, for they are called so by the Egyptians. Though Seth was an Egyptian god, as god of the Hyksos he represents some Asiatic deity. The possibility of a connexion between the Hyksos and the Israelites is still admitted in some quarters. Hatred of these impious foreigners, of which there is some trace in more than one text, aroused amongst the Egyptians (as nothing ever did before or since) that martial spirit which carried the armies of Tethmosis to the Euphrates.
Besides the histories of Egypt, see J. H. Breasted, _Ancient Records of Egypt_; Historical Documents ii. 4, 125; G. Maspero, _Contes populaires_, 3me éd. p. 236; W. M. F. Petrie, _Hyksos and Israelite Cities_, p. 67; Golénischeff in _Recueil de travaux_, xv. p. 131. (F. Ll. G.)
HYLAS, In Greek legend, son of Theiodamas, king of the Dryopians in Thessaly, the favourite of Heracles and his companion on the Argonautic expedition. Having gone ashore at Kios in Mysia to fetch water, he was carried off by the nymphs of the spring in which he dipped his pitcher. Heracles sought him in vain, and the answer of Hylas to his thrice-repeated cry was lost in the depths of the water. Ever afterwards, in memory of the threat of Heracles to ravage the land if Hylas were not found, the inhabitants of Kios every year on a stated day roamed the mountains, shouting aloud for Hylas (Apollonius Rhodius i. 1207; Theocritus xiii.; Strabo xii. 564; Propertius i. 20; Virgil, _Ecl._ vi. 43). But, although the legend is first told in Alexandrian times, the "cry of Hylas" occurs long before as the "Mysian cry" in Aeschylus (_Persae_, 1054), and in Aristophanes (_Plutus_, 1127) "to cry Hylas" is used proverbially of seeking something in vain. Hylas, like Adonis and Hyacinthus, represents the fresh vegetation of spring, or the water of a fountain, which dries up under the heat of summer. It is suggested that Hylas was a harvest deity and that the ceremony gone through by the Kians was a harvest festival, at which the figure of a boy was thrown into the water, signifying the dying vegetation-spirit of the year.
See G. Türk in _Breslauer Philologische Abhandlungen_, vii. (1895); W. Mannhardt, _Mythologische Forschungen_ (1884).
HYLOZOISM (Gr. [Greek: hylê], matter, [Greek: zôê], life), in philosophy, a term applied to any system which explains all life, whether physical or mental, as ultimately derived from matter ("cosmic matter," _Weldstoff_). Such a view of existence has been common throughout the history of thought, and especially among physical scientists. Thus the Ionian school of philosophy, which began with Thales, sought for the beginning of all things in various material substances, water, air, fire (see IONIAN SCHOOL). These substances were regarded as being in some sense alive, and taking some active part in the development of being. This primitive hylozoism reappeared in modified forms in medieval and Renaissance thought, and in modern times the doctrine of materialistic monism is its representative. Between modern materialism and hylozoism proper there is, however, the distinction that the ancients, however vaguely, conceived the elemental matter as being in some sense animate if not actually conscious and conative.
HYMEN, or HYMENAEUS, originally the name of the song sung at marriages among the Greeks. As usual the name gradually produced the idea of an actual person whose adventures gave rise to the custom of this song. He occurs often in association with Linus and Ialemus, who represent similar personifications, and is generally called a son of Apollo and a Muse. As the son of Dionysus and Aphrodite, he was regarded as a god of fruitfulness. In Attic legend he was a beautiful youth who, being in love with a girl, followed her in a procession to Eleusis disguised as a woman, and saved the whole band from pirates. As reward he obtained the girl in marriage, and his happy married life caused him ever afterwards to be invoked in marriage songs (Servius on Virgil, _Aen._ i. 651). According to another story, he was a youth who was killed by the fall of his house on his wedding day; hence he was invoked, to propitiate him and avert a similar fate from others (Servius, loc. cit.). He is represented in works of art as an effeminate-looking, winged youth, carrying a bridal torch and wearing a nuptial veil. The marriage song was sung, with musical accompaniment, during the procession of the bride from her parents' house to that of the bridegroom, Hymenaeus being invoked at the end of each portion.
See R. Schmidt, _De Hymenaeo el Talasio_ (1886), and J. A. Hild in Daremberg and Saglis's _Dictionnaire des antiquités_.
HYMENOPTERA (Gr. [Greek: hymên], a membrane, and [Greek: pteron], a wing), a term used in zoological classification for one of the most important orders of the class _Hexapoda_ (q.v.). The order was founded by Linnaeus (_Systema Naturae_, 1735), and is still recognized by all naturalists in the sense proposed by him, to include the saw-flies, gall-flies, ichneumon-flies and their allies, ants, wasps and bees. The relationship of the Hymenoptera to other orders of insects is discussed in the article HEXAPODA, but it may be mentioned here that in structure the highest members of the order are remarkably specialized, and that in the perfection of their instincts they stand at the head of all insects and indeed of all invertebrate animals. About 30,000 species of Hymenoptera are now known.
_Characters._--In all Hymenoptera the mandibles (fig. 1, C, D) are well developed, being adapted, as in the more lowly winged insects, such as the Orthoptera, for biting. The more generalized Hymenoptera have the second maxillae but slightly modified, their inner lobes being fused to form a _ligula_ (fig. 1, B, b). In the higher families this structure becomes elongated (fig. 2, g) so as to form an elaborate sucking-organ or "tongue." These insects are able, therefore, to bite as well as to suck, whereas most insects which have acquired the power of suction have lost that of biting. Both fore- and hind-wings are usually present, both pairs being membranous, the hind-wings small and not folded when at rest, each provided along the costa with a row of curved hooks which catch on to a fold along the dorsum of the adjacent fore-wing during flight. A large number of Hymenoptera are, however, entirely wingless--at least as regards one sex or form of the species. One of the most remarkable features is the close union of the foremost abdominal segment (fig. 3, i.) with the metathorax, of which it often seems to form a part, the apparent first abdominal segment being, in such case, really the second (fig. 3, ii.). The true first segment, which undergoes a more or less complete fusion with the thorax is known as the "median segment" or _propodeum_. In female Hymenoptera the typical insectan ovipositor with its three pairs of processes is well developed, and in the higher families this organ becomes functional as a sting (fig. 5),--used for offence and defence. As regards their life history, all Hymenoptera undergo a "complete" metamorphosis. The larva is soft-skinned (eruciform), being either a caterpillar (fig. 6, b) or a legless grub (fig. 7, a), and the pupa is free (fig. 7, c), i.e. with the appendages not fixed to the body, as is the case in the pupa of most moths.
_Structure._--The head of a hymenopterous insect bears three simple eyes (ocelli) on the front and vertex in addition to the large compound eyes. The feelers are generally simple in type, rarely showing serrations or prominent appendages; but one or two basal segments are frequently differentiated to form an elongate "scape," the remaining segments--carried at an elbowed angle to the scape--making up the "flagellum"; the segments of the flagellum often bear complex sensory organs. The general characters of the jaws have been mentioned above, and in detail there is great variation in these organs among the different families. The sucking tongue of the Hymenoptera has often been compared with the hypopharynx of other insects. According to D. Sharp, however, the hypopharynx is present in all Hymenoptera as a distinct structure at the base of the "tongue," which must be regarded as representing the fused laciniae of the second maxillae. In the thorax the pronotum and prosternum are closely associated with the mesothorax, but the pleura of the prothorax are usually shifted far forwards, so that the fore-legs are inserted just behind the head. A pair of small plates--the tegulae--are very generally present at the bases of the fore-wings. The union of the first abdominal segment with the metathorax has been already mentioned. The second (so-called "first") abdominal segment is often very constricted, forming the "waist" so characteristic of wasps and ants for example. The constriction of this segment and its very perfect articulation with the propodeum give great mobility to the abdomen, so that the ovipositor or sting can be used with the greatest possible accuracy and effect.
Mention has already been made of the series of curved hooks along the costa of the hind-wing; by means of this arrangement the two wings of a side are firmly joined together during flight, which thus becomes particularly accurate. The wings in the Hymenoptera show a marked reduction in the number of nervures as compared with more primitive insects. The main median nervure, and usually also the sub-costal become united with the radial, while the branches of radial, median and cubital nervures pursuing a transverse or recurrent course across the wing, divide its area into a number of areolets or "cells," that are of importance in classification. Among many of the smaller Hymenoptera we find that the wings are almost destitute of nervures. In the hind-wings--on account of their reduced size--the nervures are even more reduced than in the fore-wings.
The legs of Hymenoptera are of the typical insectan form, and the foot is usually composed of five segments. In many families the trochanter appears to be represented by two small segments, there being thus an extra joint in the leg. It is almost certain that the distal of these two segments really belongs to the thigh, but the ordinary nomenclature will be used in the present article, as this character is of great importance in discriminating families, and the two segments in question are referred to the trochanter by most systematic writers.
The typical insectan ovipositor, so well developed among the Hymenoptera, consists of three pairs of processes (gonapophyses) two of which belong to the ninth abdominal segment and one to the eighth. The latter are the cutting or piercing stylets (fig. 5, A) of the ovipositor, while the two outer processes of the ninth segment are modified into sheaths or feelers (fig. 5, C) and the two inner processes form a guide (fig. 5, B) on which the stylets work, tongues or rails on the "guide" fitting accurately into longitudinal grooves on the stylet. In the different families of the Hymenoptera, there are various modifications of the ovipositor, in accord with the habits of the insects and the purposes to which the organ is put. The sting of wasps, ants and bees is a modified ovipositor and is used for egg-laying by the fertile females, as well as for defence. Most male Hymenoptera have processes which form claspers or genital armature. These processes are not altogether homologous with those of the ovipositor, being formed by inner and outer lobes of a pair of structures on the ninth abdominal segment.
Many points of interest are to be noted in the internal structure of the Hymenoptera. The gullet leads into a moderate-sized crop, and several pairs of salivary glands open into the mouth. The crop is followed by a proventriculus which, in the higher Hymenoptera, forms the so-called "honey stomach," by the contraction of whose wails the solid and liquid food can be separated, passed on into the digestive stomach, or held in the crop ready for regurgitation into the mouth. Behind the digestive stomach are situated, as usual, intestine and rectum, and the number of kidney (Malpighian) tubes varies from only six to over a hundred, being usually great.
In the female, each ovary consists of a large number of ovarian tubes, in which swollen chambers containing the egg-cells alternate with smaller chambers enclosing nutrient material. In connexion with the ovipositor are two poison-glands, one acid and the other alkaline in its secretion. The acid gland consists of one, two or more tubes, with a cellular coat of several layers, opening into a reservoir whence the duct leads to the exterior. The alkaline gland is an irregular tube with a single cellular layer, its duct opening alongside that of the acid reservoir. These glands are most strongly developed when the ovipositor is modified into a sting.
_Development._--Parthenogenesis is of normal occurrence in the life-cycle of many Hymenoptera. There are species of gall-fly in which males are unknown, the unfertilized eggs always developing into females. On the other hand, in certain saw-flies and among the higher families, the unfertilized eggs, capable of development, usually give rise to male insects (see BEE). The larvae of most saw-flies feeding on the leaves of plants are caterpillars (fig. 6, b) with numerous abdominal pro-legs, but in most families of Hymenoptera the egg is laid in such a situation that an abundant food-supply is assured without exertion on the part of the larva, which is consequently a legless grub, usually white in colour, and with soft flexible cuticle (fig. 7, a). The organs and instincts for egg-laying and food-providing are perhaps the most remarkable features in the economy of the Hymenoptera. Gall-fly grubs are provided with vegetable food through the eggs being laid by the mother insect within plant tissues. The ichneumon pierces the body of a caterpillar and lays her eggs where the grubs will find abundant animal food. A digging-wasp hunts for insect prey and buries it with the egg, while a true wasp feeds her brood with captured insects, as a bird her fledglings. Bees store honey and pollen to serve as food for their young. Thus we find throughout the order a degree of care for offspring unreached by other insects, and this family-life has, in the best known of the Hymenoptera--ants, wasps and bees--developed into an elaborate social organization.
_Social Life._--The development of a true insect society among the Hymenoptera is dependent on a differentiation among the females between individuals with well-developed ovaries ("queens") whose special function is reproduction; and individuals with reduced or aborted ovaries ("workers") whose duty is to build the nest, to gather food and to tend and feed the larvae. Among the wasps the workers may only differ from the queens in size, and individuals intermediate between the two forms of female may be met with. Further, the queen wasp, and also the queen humble-bee, commences unaided the work of building and founding a new nest, being afterwards helped by her daughters (the workers) when these have been developed. In the hive-bee and among ants, on the other hand, there are constant structural distinctions between queen and worker, and the function of the queen bee in a hive is confined to egg-laying, the labour of the community being entirely done by the workers. Many ants possess several different forms of worker, adapted for special duties. Details of this fascinating subject are given in the special articles ANT, BEE and WASP (q.v.).
_Habits and Distribution._--Reference has been already made to the various methods of feeding practised by Hymenoptera in the larval stage, and the care taken of or for the young throughout the order leads in many cases to the gathering of such food by the mother or nurse. Thus, wasps catch flies; worker ants make raids and carry off weak insects of many kinds; bees gather nectar from flowers and transform it into honey within their stomachs--largely for the sake of feeding the larvae in the nest. The feeding habits of the adult may agree with that of the larva, or differ, as in the ease of wasps which feed their grubs on flies, but eat principally vegetable food themselves. The nest-building habit is similarly variable. Digging wasps make simple holes in the ground; many burrowing bees form branching tunnels; other bees excavate timber or make their brood-chambers in hollow plant-stems; wasps work up with their saliva vegetable fibres bitten off tree-bark to make paper; social bees produce from glands in their own bodies the wax whence their nest-chambers are built. The inquiline habit ("cuckoo-parasitism"), when one species makes use of the labour of another by invading the nest and laying her eggs there, is of frequent occurrence among Hymenoptera; and in some cases the larva of the intruder is not content with taking the store of food provided, but attacks and devours the larva of the host.
Most Hymenoptera are of moderate or small size, the giants of the order--certain saw-flies and tropical digging-wasps--never reach the bulk attained by the largest beetles, while the wing-spread is narrow compared with that of many dragon-flies and moths. On the other hand, there are thousands of very small species, and the tiny "fairy-flies" (_Mymaridae_), whose larvae live as parasites in the eggs of various insects, are excessively minute for creatures of such complex organization. Hymenoptera are probably less widely distributed than Aptera, Coleoptera or Diptera, but they are to be found in all except the most inhospitable regions of the globe. The order is, with few exceptions, terrestrial or aerial in habit. Comparatively only a few species are, for part of their lives, denizens of fresh water; these, as larvae, are parasitic on the eggs or larvae of other aquatic insects, the little hymenopteron, _Polynema natans_, one of the "fairy-flies"--swims through the water by strokes of her delicate wings in search of a dragon-fly's egg in which to lay her own egg, while the rare _Agriotypus_ dives after the case of a caddis-worm. It is of interest that the waters have been invaded by the parasitic group of the Hymenoptera, since in number of species this is by far the largest of the order. No group of terrestrial insects escapes their attacks--even larvae boring in wood are detected by ichneumon flies with excessively long ovipositors. Not a few cases are known in which a parasitic larva is itself pierced by the ovipositor of a "hyperparasite," and even the offspring of the latter may itself fall a victim to the attack of a "tertiary parasite."
_Fossil History._--Very little is known of the history of the Hymenoptera previous to the Tertiary epoch, early in which, as we know from the evidence of many Oligocene and Miocene fossils, all the more important families had been differentiated. Fragments of wings from the Lias and Oolitic beds have been referred to ants and bees, but the true nature of these remains is doubtful.
_Classification._--Linnaeus divided the Hymenoptera into two sections--the Terebrantia, whose females possess a cutting or piercing ovipositor, and the Aculeata, in which the female organ is modified into a sting. This nomenclature was adopted by P. A. Latreille and has been in general use until the present day. A closely similar division of the order results from T. Hartig's character drawn from the trochanter--whether of two segments or undivided--the groups being termed respectively Ditrocha and Monotrocha. But the most natural division is obtained by the separation of the saw-flies as a primitive sub-order, characterized by the imperfect union of the first abdominal segment with the thorax, and by the broad base of the abdomen, so that there is no median constriction or "waist," and by the presence of thoracic legs--usually also of abdominal pro-legs--in the larva. All the other families of Hymenoptera, including the gall-flies, ichneumons and aculeates, have the first abdominal segment closely united with the thorax, the second abdominal segment constricted so as to form a narrow stalk or "waist," and legless larvae without a hinder outlet to the food-canal. These two sub-orders are usually known as the _Sessiliventra_ and _Petioliventra_ respectively, but the names _Symphyta_ and _Apocrita_ proposed in 1867 by C. Gerstaecker have priority, and should not be replaced.
_Symphyta._
This sub-order, characterized by the "sessile," broad-based abdomen, whose first segment is imperfectly united with the thorax, and by the usually caterpillar-like larvae with legs, includes the various groups of saw-flies. Three leading families may be mentioned. The _Cephidae_, or stem saw-flies, have an elongate pronotum, a compressed abdomen, and a single spine on the shin of the fore-leg. The soft, white larvae have the thoracic legs very small and feed in the stems of various plants. _Cephus pygmaeus_ is a well-known enemy of corn crops. The _Siricidae_ ("wood-wasps") are large elongate insects also with one spine on each fore-shin, but with the pronotum closely joined to the mesothorax. The ovipositor is long and prominent, enabling the female insect to lay her eggs in the wood of trees, where the white larvae, whose legs are excessively short, tunnel and feed. These insects are adorned with bands of black and yellow, or with bright metallic colours, and on account of their large size and formidable ovipositors they often cause needless alarm to persons unfamiliar with their habits. The _Tenthredinidae_, or true saw-flies, are distinguished by two spines on each fore-shin, while the larvae are usually caterpillars, with three pairs of thoracic legs, and from six to eight pairs of abdominal pro-legs the latter not possessing the hooks found on the pro-legs of lepidopterous caterpillars. Most saw-fly larvae devour leaves, and the beautifully serrate processes of the ovipositor are well adapted for egg-laying in plant tissues. Some saw-fly larvae are protected by a slimy secretion (fig. 6, c) and a few live concealed in galls. In the form of the feelers, the wing-neuration and minor structural details there is much diversity among the saw-flies. They have been usually regarded as a single family, but W. H. Ashmead has lately differentiated eleven families of them.
_Apocrita._
This sub-order includes the vast majority of the Hymenoptera, characterized by the narrowly constricted waist in the adult and by the legless condition of the larva. The trochanter is simple in some genera and divided in others. With regard to the minor divisions of this group, great difference of opinion has prevailed among students. In his recent classification Ashmead (1901) recognizes seventy-nine families arranged under eight "super-families." The number of species included in this division is enormous, and the multiplication of families is, to some extent, a natural result of increasingly close study. But the distinctions between many of these rest on comparatively slight characters, and it is likely that the future discovery of new genera may abolish many among such distinctions as may now be drawn. It seems advisable, therefore, in the present article to retain the wider conception of the family that has hitherto contented most writers on the Hymenoptera. Ashmead's "super-families" have, however, been adopted as--founded on definite structural characters--they probably indicate relationship more nearly than the older divisions founded mostly on habit. The Cynipoidea include the gall-flies and their parasitic relations. In the Chalcidoidea, Ichneumonoidea and Proctotrypoidea will be found nearly all the "parasitic Hymenoptera" of older classifications. The Formicoidea are the ants. The group of Fossores, or "digging-wasps," is divided by Ashmead, one section forming the Sphecoidea, while the other, together with the Chrysidae and the true wasps, make up the Vespoidea. The Apoidea consists of the bees only.
_Cynipoidea._--In this division the ovipositor issues from the ventral surface of the abdomen; the pronotum reaches back to the tegulae; the trochanter has two segments; the fore-wing (fig. 4, 2) has no stigma, but one or two areolets. The feelers with twelve to fifteen segments are thread-like and straight. All the insects included in this group are small and form two families--the Cynipidae and the Figitidae. They are the "gall-flies," many of the species laying eggs in various plant-tissues where the presence of the larva causes the formation of a pathological growth or gall, always of a definite form and characteristic of the species; the "oak-apple" and the bedeguar of the rose are familiar examples. Other flies of this group have the inquiline habit, laying their eggs in the galls of other species, while others again pierce the cuticle of maggots or aphids, in whose bodies their larvae live as parasites.
_Chalcidoidea._--This division resembles the Cynipoidea in the position of the ovipositor, and in the two segmented trochanters. The fore-wing also has no stigma, and the whole wing is almost destitute of nervures and areolets, while the pronotum does not reach back to the tegulae, and the feelers are elbowed (fig. 7). The vast majority of this group, including nearly 5000 known species, are usually reckoned as a single family, the _Chalcididae_, comprising small insects, often of bright metallic colours, whose larvae are parasitic in insects of various orders. The "fig-insects," whose presence in ripening figs is believed essential to the proper development of the fruit, belong to _Blastophaga_ and other genera of this family. They are remarkable in having wingless males and winged females. The "polyembryonic" development of an _Encyrtus_, as studied by P. Marchal, is highly remarkable. The female lays her egg in the egg of a small ermine moth (_Hyponomeuta_) and the egg gives rise not to a single embryo but to a hundred, which develop as the host-caterpillar develops, being found at a later stage within the latter enveloped in a flexible tube.
The _Mymaridae_ or "fairy-flies" are distinguished from the _Chalcididae_ by their narrow fringed wings (figs. 4, 5) and by the situation of the ovipositor just in front of the tip of the abdomen. They are among the most minute of all insects and their larvae are probably all parasitic in insects' eggs.
_Ichneumonoidea._--The ten thousand known species included in this group agree with the Cynipoidea and Chalcidoidea in the position of the ovipositor and in the jointed trochanters, but are distinguished by the fore-wing possessing a distinct stigma and usually a typical series of nervures and areolets (figs. 4, 8). Many of the species are of fair size. They lay their eggs (fig. 8) in the bodies of insects and their larvae belonging to various orders. A few small families such as the _Evaniidae_ and the _Stephanidae_ are included here, but the vast majority of the group fall into two large families, the _Ichneumonidae_ and the _Braconidae_, the former distinguished by the presence of two median (or discoidal) cells in the fore-wing (figs. 4, 7), while the latter has only one (figs. 4, 6). Not a few of these insects, however, are entirely wingless. On account of their work in destroying plant-eating insects, the ichneumon-flies are of great economic importance.
_Proctotrypoidea._--This group may be distinguished from the preceding by the position of the ovipositor at the extreme apex of the abdomen, and from the groups that follow (with very few exceptions) by the jointed trochanters of the legs. The pronotum reaches back to the tegulae. The _Pelecinidae_--included here by Ashmead--are large insects with remarkably elongate abdomens and undivided trochanters. All the other members of the group may be regarded as forming a single family--the _Proctotrypidae_, including an immense number of small parasitic Hymenoptera, not a few of which are wingless. Of special interest are the transformations of _Platygaster_, belonging to this family, discovered by M. Ganin, and familiarized to English readers through the writings of Sir J. Lubbock (Lord Avebury). The first larva is broad in front and tapers behind to a "tail" provided with two divergent processes, so that it resembles a small crustacean. It lives in the grub of a gall-midge and it ultimately becomes changed into the usual white and fleshy hymenopterous larva. The four succeeding sections, in which the ovipositor is modified into a sting (always exserted from the tip of the abdomen) and the trochanters are with few exceptions simple, form the _Aculeata_ of Linnaeus.
_Formicoidea._--The ants which form this group are readily distinguished by the differentiation of the females into winged "queens" and wingless "workers." The pronotum extends back to the wing-bases, and the "waist" is greatly constricted and marked by one or two "nodes." The differentiation of the females leads to a complex social life, the nesting habits of ants and the various industries that they pursue being of surpassing interest (see ANT).
_Vespoidea._--This section includes a number of families characterized by the backward extension of the prothorax to the tegulae and distinguished from the ants by the absence of "nodes" at the base of the abdomen. The true wasps have the fore-wings folded lengthwise when at rest and the fore-legs of normal build--not specialized for digging. The _Vespidae_ or social wasps have "queens" and "workers" like the ants, but both these forms of female are winged; the claws on their fret are simple. In the _Eumenidae_ or solitary wasps the female sex is undifferentiated, and the foot claws are toothed. (For the habits of these insects see WASP.) The _Chrysididae_ or ruby wasps are small insects with a very hard cuticle exhibiting brilliant metallic colours--blue, green and crimson. Only three or four abdominal segments are visible, the hinder segments being slender and retracted to form a telescope-like tube in which the ovipositor lies. When the ovipositor is brought into use this tube is thrust out. The eggs are laid in the nests of various bees and wasps, the chrysid larva living as a "cuckoo" parasite. The _Trigonalidae_, a small family whose larvae are parasitic in wasps' nests, also probably belong here.
The other families of the _Vespoidea_ belong to the series of "Fossores" or digging-wasps. In two of the families--the _Mutillidae_ and _Thynnidae_--the females are wingless and the larvae live as parasites in the larvae of other insects; the female _Mutilla_ enters bumble-bees' nests and lays her eggs in the bee-grubs. In the other families both sexes are winged, and the instinct and industry of the females are among the most wonderful in the Hymenoptera. They make burrows wherein they place insects or spiders which they have caught and stung, laying their eggs beside the victim so that the young larvae find themselves in presence of an abundant and appropriate food-supply. Valuable observations on the habits of these insects are due to J. H. Fabre and G. W. and E. Peckham. The prey is sometimes stung in the neighbourhood of the nerve ganglia, so that it is paralysed but not killed, the grub of the fossorial wasp devouring its victim alive; but this instinct varies in perfection, and in many cases the larva flourishes equally whether its prey be killed or not. The females have a wonderful power of finding their burrows on returning from their hunting expeditions. Among the Vespoid families of fossorial wasps, the _Pompilidae_ are the most important. They are recognizable by their slender and elongate hind-legs; many of them provision their burrows with spiders. The _Sapygidae_ are parasitic on bees, while the _Scoliidae_ are large, robust and hairy insects, many of which prey upon the grubs of chafers.
_Sphecoidea._--In this division are included the rest of the "digging-wasps," distinguished from the _Vespoidea_ by the short pronotum not reaching backward to the tegulae. They have usually been reckoned as forming a single, very large family--the _Sphegidae_--but ten or twelve subdivisions of the group are regarded as distinct families by Ashmead and others. Great diversity is shown in the details of structure, habits and nature of the prey. Species of _Sphex_, studied by Fabre, provisioned their brood-chambers with crickets. _Pelopoeus_ hunts spiders, while _Ammophila_ catches caterpillars for the benefit of her young. Fabre states that the last-named insect uses a stone for the temporary closing of her burrow, and the Peckhams have seen a female _Ammophila_ take a stone between her mandibles and use it as a hammer for pounding down the earth over her finished nest. The habits of _Bembex_ are of especial interest. The female, instead of provisioning her burrow with a supply of food that will suffice the larva for its whole life, brings fresh flies with which she regularly feeds her young. In this instinct we have a correspondence with the habits of social wasps and bees. Yet it may be thought that the usual instinct of the "digging-wasps" to capture and store up food in an underground burrow for the benefit of offspring which they will never see is even more surprising. The habit of some genera is to catch the prey before making their tunnel, but more frequently the insect digs her nest, and then hunts for prey to put into it.
_Apoidea._--The bees which make up this group agree with the Sphecoidea in the short pronotum, but may be distinguished from all other Hymenoptera by the widened first tarsal segment and the plumose hairs on head and body. They are usually regarded as forming a single family--the _Apidae_--but there is very great diversity in structural details, and Ashmead divides them into fourteen families. The "tongue," for example, is short and obtuse or emarginate in _Colletes_ and _Prosopis_, while in all other bees it is pointed at the tip. But in _Andrena_ and its allies it is comparatively short, while in the higher genera, such as _Apis_ and _Bombus_, it is elongate and flexible, forming a most elaborate and perfect organ for taking liquid food. Bees feed on honey and pollen. Most of the genera are "solitary" in habit, the female sex being undifferentiated; but among the humble-bees and hive-bees we find, as in social wasps and ants, the occurrence of workers, and the consequent elaboration of a wonderful insect-society. (See BEE.)
BIBLIOGRAPHY.--The literature of several special families of the Hymenoptera will be found under the articles ANT, BEE, ICHNEUMON-FLY, WASP, &c., referred to above. Among earlier students on structure may be mentioned P. A. Latreille, _Familles naturelles du règne animal_ (Paris, 1825), who recognized the nature of the "median segment." C. Gerstaecker (_Arch. f. Naturg._ xx., 1867) and F. Brauer (_Sitzb. K. Akad. Wiss. Wien._ lxxxv., 1883) should also be consulted on this subject. For internal anatomy, specially the digestive organs, see L. Dufour, _Mém. savants étrangers_, vii. (1841), and _Ann. Sci. Nat. Zool._ (4), i. 1854. For nervous system H. Viallanes, _Ann. Sci. Nat. Zool._ (7), ii. iv. 1886-1887, and F. C. Kenyon, _Journ. Comp. Neurol._ vi., 1896. For poison and other glands, see L. Bordas, _Ann. Sci. Nat. Zool._ (7) xix., 1895. For the sting and ovipositor H. Dewitz, _Zeits. wiss. Zool._ xxv., 1874, xxviii., 1877, and F. Zander, _ib._ lxvi., 1899. For male genital armature S. A. Peytoureau, _Morphologie de l'armure génitale des insectes_ (Bordeaux, 1895), and E. Zander, Zeits. wiss. Zool. lxvii., 1900. The systematic student of Hymenoptera is greatly helped by C. G. de Dalla Torre's _Catalogus Hymenopterorum_ (10 vols., Leipzig, 1893-1902). For general classifications see F. W. Konow, _Entom. Nachtr._ (1897), and W. H. Ashmead, _Proc. U.S. Nat. Mus._ xxiii., 1901; the latter paper deals also especially with the Ichneumonoidea of the globe. For habits and life histories of Hymenoptera see J. Lubbock (Lord Avebury), _Ants, Bees and Wasps_ (9th ed., London, 1889); C. Janet, _Études sur les fourmis, les guêpes et les abeilles_ (Paris, &c., 1893 and onwards); and G. W. and E. G. Peckham, _Instincts and Habits of Solitary Wasps_ (Madison, Wis. U.S.A., 1898). Monographs of most of the families of British Hymenoptera have now been published. For saw-flies and gall-flies, see P. Cameron's _British Phytophagous Hymenoptera_ (4 vols., London, _Roy. Soc._, 1882-1893). For Ichneumonoidea, C. Morley's _Ichneumons of Great Britain_ (Plymouth, 1903, &c.), and T. A. Marshall's "British Braconidae," _Trans. Entom. Soc._, 1885-1899. The smaller parasitic Hymenoptera have been neglected in this country since A. H. Haliday's classical papers _Entom. Mag._ i.-v., (1833-1838) but Ashmead's "North American Proctotrypidae" (_Bull. U.S. Nat. Mus._ xlv., 1893) is valuable for the European student. For the Fossores, wasps, ants and bees see E. Saunders, _Hymenoptera Aculeata of the British Islands_ (London, 1896). Exhaustive references to general systematic works will be found in de Dalla Torre's _Catalogue_ mentioned above. Of special value to English students are C. T. Bingham's _Fauna of British India_, "Hymenoptera" (London, 1897 and onwards), and P. Cameron's volumes on Hymenoptera in the _Biologia Centrali-Americana_. F. Smith's _Catalogues of Hymenoptera in the British Museum_ (London, 1853-1859) are well worthy of study. (G. H. C.)
HYMETTUS (Ital. Monte Matto, hence the modern name Trello Vouni), a mountain in Attica, bounding the Athenian plain on the S.E. Height, 3370 ft. It was famous in ancient times for its bees, which gathered honey of peculiar flavour from its aromatic herbs; their fame still persists. The spring mentioned by Ovid (_Ars Amat._ iii. 687) is probably to be recognized near the monastery of Syriani or Kaesariani on the western slope. This may be identical with that known as [Greek: Kyllon Pêra], said to be a remedy for barrenness in women. The marble of Hymettus, which often has a bluish tinge, was used extensively for building in ancient Athens, and also, in early times, for sculpture; but the white marble of Pentelicus was preferred for both purposes.
See E. Dodwell, _Classical and Topographical Tour_ (1819), i. 483.
HYMNS.--1. _Classical Hymnody._--The word "hymn" ([Greek: hymnos]) was employed by the ancient Greeks[1] to signify a song or poem composed in honour of gods, heroes or famous men, or to be recited on some joyful, mournful or solemn occasion. Polymnia was the name of their lyric muse. Homer makes Alcinous entertain Odysseus with a "hymn" of the minstrel Demodocus, on the capture of Troy by the wooden horse. The _Works and Days_ of Hesiod begins with an invocation to the Muses to address hymns to Zeus, and in his _Theogonia_ he speaks of them as singing or inspiring "hymns" to all the divinities, and of the bard as "their servant, hymning the glories of men of old, and of the gods of Olympus." Pindar calls by this name odes, like his own, in praise of conquerors at the public games of Greece. The Athenian dramatists (Euripides most frequently) use the word and its cognate verbs in a similar manner; they also describe by them metrical oracles and apophthegms, martial, festal and hymeneal songs, dirges and lamentations or incantations of woe.
Hellenic hymns, according to this conception of them, have come down to us, some from a very early and others from a late period of Greek classical literature. Those which passed by the name of Homer[2] were already old in the time of Thucydides. They are mythological poems (several of them long), in hexameter verse--some very interesting. That to Apollo contains a traditionary history of the origin and progress of the Delphic worship; those on Hermes and on Dionysus are marked by much liveliness and poetical fancy. Hymns of a like general character, but of less interest (though these also embody some fine poetical traditions of the Greek mythology, such as the story of Teiresias, and that of the wanderings of Leto), were written in the 3rd century before Christ, by Callimachus of Cyrene. Cleanthes, the successor of Zeno, composed (also in hexameters) an "excellent and devout hymn" (as it is justly called by Cudworth, in his _Intellectual System_) to Zeus, which is preserved in the _Eclogae_ of Stobaeus, and from which Aratus borrowed the words, "For we are also His offspring," quoted by St Paul at Athens. The so-called Orphic hymns, in hexameter verse, styled [Greek: teletai], or hymns of initiation into the "mysteries" of the Hellenic religion, are productions of the Alexandrian school,--as to which learned men are not agreed whether they are earlier or later than the Christian era.
The Romans did not adopt the word "hymn"; nor have we many Latin poems of the classical age to which it can properly be applied. There are, however, a few--such as the simple and graceful "Dianae sumus in fide" ("Dian's votaries are we") of Catullus, and "Dianam tenerae dicite virgines" ("Sing to Dian, gentle maidens") of Horace--which approach much more nearly than anything Hellenic to the form and character of modern hymnody.
2. _Hebrew Hymnody._--For the origin and idea of Christian hymnody we must look, not to Gentile, but to Hebrew sources. St Augustine's definition of a hymn, generally accepted by Christian antiquity, may be summed up in the words, "praise to God with song" ("cum cantico"); Bede understood the "canticum" as properly requiring metre; though he thought that what in its original language was a true hymn might retain that character in an unmetrical translation. Modern use has enlarged the definition; Roman Catholic writers extend it to the praises of saints; and the word now comprehends rhythmical prose as well as verse, and prayer and spiritual meditation as well as praise.
The modern distinction between psalms and hymns is arbitrary (see PSALMS). The former word was used by the LXX. as a generic designation, probably because it implied an accompaniment by the psaltery (said by Eusebius to have been of very ancient use in the East) or other instruments. The cognate verb "psallere" has been constantly applied to hymns, both in the Eastern and in the Western Church; and the same compositions which they described generically as "psalms" were also called by the LXX. "odes" (i.e. songs) and "hymns." The latter word occurs, e.g. in Ps. lxxii. 20 ("the hymns of David the son of Jesse"), in Ps. lxv. 1, and also in the Greek titles of the 6th, 54th, 55th, 67th and 76th (this numbering of the psalms being that of the English version, not of the LXX.). The 44th chapter of Ecclesiasticus, "Let us now praise famous men," &c., is entitled in the Greek [Greek: paterôn hymnos], "The Fathers' Hymn." Bede speaks of the whole book of Psalms as called "liber hymnorum," by the universal consent of Hebrews, Greeks and Latins.
In the New Testament we find our Lord and His apostles singing a hymn ([Greek: hymnêsantes exêlthon]), after the institution of the Lord's Supper; St Paul and Silas doing the same ([Greek: hymnoun ton theon]) in their prison at Philippi; St James recommending psalm-singing ([Greek: psalletô]), and St Paul "psalms and hymns and spiritual songs" ([Greek: psalmois kai hymnois kai ôdais pneumatikais]) St Paul also, in the 14th chapter of the first epistle to the Corinthians, speaks of singing ([Greek: psalô]) and of every man's psalm ([Greek: hekastos hymôn psalmon echei]). In a context which plainly has reference to the assemblies of the Corinthian Christians for common worship. All the words thus used were applied by the LXX. to the Davidical psalms; it is therefore possible that these only may be intended, in the different places to which we have referred. But there are in St Paul's epistles several passages (Eph. v. 14; 1 Tim. iii. 16; 1 Tim. vi. 15, 16; 2 Tim.