PART I.--STATICS

§ 1. _Statics of a Particle._--By a _particle_ is meant a body whose position can for the purpose in hand be sufficiently specified by a mathematical point. It need not be "infinitely small," or even small compared with ordinary standards; thus in astronomy such vast bodies as the sun, the earth, and the other planets can for many purposes be treated merely as points endowed with mass.

A _force_ is conceived as an effort having a certain direction and a certain magnitude. It is therefore adequately represented, for mathematical purposes, by a straight line AB drawn in the direction in question, of length proportional (on any convenient scale) to the magnitude of the force. In other words, a force is mathematically of the nature of a "vector" (see VECTOR ANALYSIS, QUATERNIONS). In most questions of pure statics we are concerned only with the _ratios_ of the various forces which enter into the problem, so that it is indifferent what _unit_ of force is adopted. For many purposes a gravitational system of measurement is most natural; thus we speak of a force of so many pounds or so many kilogrammes. The "absolute" system of measurement will be referred to below in PART II., KINETICS. It is to be remembered that all "force" is of the nature of a push or a pull, and that according to the accepted terminology of modern mechanics such phrases as "force of inertia," "accelerating force," "moving force," once classical, are proscribed. This rigorous limitation of the meaning of the word is of comparatively recent origin, and it is perhaps to be regretted that some more technical term has not been devised, but the convention must now be regarded as established.

The fundamental postulate of this part of our subject is that the two forces acting on a particle may be compounded by the "parallelogram rule." Thus, if the two forces P,Q be represented by the lines OA, OB, they can be replaced by a single force R represented by the diagonal OC of the parallelogram determined by OA, OB. This is of course a physical assumption whose propriety is justified solely by experience. We shall see later that it is implied in Newton's statement of his Second Law of motion. In modern language, forces are compounded by "vector-addition"; thus, if we draw in succession vectors [->HK], [->KL] to represent P, Q, the force R is represented by the vector [->HL] which is the "geometric sum" of [->HK], [->KL].

By successive applications of the above rule any number of forces acting on a particle may be replaced by a single force which is the vector-sum of the given forces: this single force is called the _resultant_. Thus if [->AB], [->BC], [->CD] ..., [->HK] be vectors representing the given forces, the resultant will be given by [->AK]. It will be understood that the figure ABCD ... K need not be confined to one plane.

If, in particular, the point K coincides with A, so that the resultant vanishes, the given system of forces is said to be in _equilibrium_--i.e. the particle could remain permanently at rest under its action. This is the proposition known as the _polygon of forces_. In the particular case of three forces it reduces to the _triangle of forces_, viz. "If three forces acting on a particle are represented as to magnitude and direction by the sides of a triangle taken in order, they are in equilibrium."

A sort of converse proposition is frequently useful, viz. if three forces acting on a particle be in equilibrium, and any triangle be constructed whose sides are respectively parallel to the forces, the magnitudes of the forces will be to one another as the corresponding sides of the triangle. This follows from the fact that all such triangles are necessarily similar.

As a simple example of the geometrical method of treating statical problems we may consider the equilibrium of a particle on a "rough" inclined plane. The usual empirical law of sliding friction is that the mutual action between two plane surfaces in contact, or between a particle and a curve or surface, cannot make with the normal an angle exceeding a certain limit [lambda] called the _angle of friction_. If the conditions of equilibrium require an obliquity greater than this, sliding will take place. The precise value of [lambda] will vary with the nature and condition of the surfaces in contact. In the case of a body simply resting on an inclined plane, the reaction must of course be vertical, for equilibrium, and the slope [alpha] of the plane must therefore not exceed [lambda]. For this reason [lambda] is also known as the _angle of repose_. If [alpha] > [lambda], a force P must be applied in order to maintain equilibrium; let [theta] be the inclination of P to the plane, as shown in the left-hand diagram. The relations between this force P, the gravity W of the body, and the reaction S of the plane are then determined by a triangle of forces HKL. Since the inclination of S to the normal cannot exceed [lambda] on either side, the value of P must lie between two limits which are represented by L1H, L2H, in the right-hand diagram. Denoting these limits by P1, P2, we have

P1/W = L1H/HK = sin ([alpha] - [lambda])/cos ([theta] + [lambda]), P2/W = L2H/HK = sin ([alpha] + [lambda])/cos ([theta] - [lambda]).

It appears, moreover, that if [theta] be varied P will be least when L1H is at right angles to KL1, in which case P1 = W sin ([alpha] - [lambda]), corresponding to [theta] = -[lambda].

Just as two or more forces can be combined into a single resultant, so a single force may be _resolved_ into _components_ acting in assigned directions. Thus a force can be uniquely resolved into two components acting in two assigned directions in the same plane with it by an inversion of the parallelogram construction of fig. 1. If, as is usually most convenient, the two assigned directions are at right angles, the two components of a force P will be P cos [theta], P sin [theta], where [theta] is the inclination of P to the direction of the former component. This leads to formulae for the analytical reduction of a system of coplanar forces acting on a particle. Adopting rectangular axes Ox, Oy, in the plane of the forces, and distinguishing the various forces of the system by suffixes, we can replace the system by two forces X, Y, in the direction of co-ordinate axes; viz.--

X = P1 cos [theta]1 + P2 cos [theta]2 + ... = [Sigma](P cos [theta]), } Y = P1 sin [theta]1 + P2 sin [theta]2 + ... = [Sigma](P sin [theta]). } (1)

These two forces X, Y, may be combined into a single resultant R making an angle [phi] with Ox, provided

X = R cos [phi], Y = R sin [phi], (2)

whence

R² = X² + Y², tan [phi] = Y/X. (3)

For equilibrium we must have R = 0, which requires X = 0, Y = 0; in words, the sum of the components of the system must be zero for each of two perpendicular directions in the plane.

A similar procedure applies to a three-dimensional system. Thus if, O being the origin, [->OH] represent any force P of the system, the planes drawn through H parallel to the co-ordinate planes will enclose with the latter a parallelepiped, and it is evident that [->OH] is the geometric sum of [->OA], [->AN], [->NH], or [->OA], [->OB], [->OC], in the figure. Hence P is equivalent to three forces Pl, Pm, Pn acting along Ox, Oy, Oz, respectively, where l, m, n, are the "direction-ratios" of [->OH]. The whole system can be reduced in this way to three forces

X = [Sigma] (Pl), Y = [Sigma] (Pm), Z = [Sigma] (Pn), (4)

acting along the co-ordinate axes. These can again be combined into a single resultant R acting in the direction ([lambda], [mu], [nu]), provided

X = R[lambda], Y = R[mu], Z = R[nu]. (5)

If the axes are rectangular, the direction-ratios become direction-cosines, so that [lambda]² + [mu]² + [nu]² = 1, whence

R² = X² + Y² + Z². (6)

The conditions of equilibrium are X = 0, Y = 0, Z = 0.

§ 2. _Statics of a System of Particles._--We assume that the mutual forces between the pairs of particles, whatever their nature, are subject to the "Law of Action and Reaction" (Newton's Third Law); i.e. the force exerted by a particle A on a particle B, and the force exerted by B on A, are equal and opposite in the line AB. The problem of determining the possible configurations of equilibrium of a system of particles subject to extraneous forces which are known functions of the positions of the particles, and to internal forces which are known functions of the distances of the pairs of particles between which they act, is in general determinate. For if n be the number of particles, the 3n conditions of equilibrium (three for each particle) are equal in number to the 3n Cartesian (or other) co-ordinates of the particles, which are to be found. If the system be subject to frictionless constraints, e.g. if some of the particles be constrained to lie on smooth surfaces, or if pairs of particles be connected by inextensible strings, then for each geometrical relation thus introduced we have an unknown reaction (e.g. the pressure of the smooth surface, or the tension of the string), so that the problem is still determinate.

The case of the _funicular polygon_ will be of use to us later. A number of particles attached at various points of a string are acted on by given extraneous forces P1, P2, P3 ... respectively. The relation between the three forces acting on any particle, viz. the extraneous force and the tensions in the two adjacent portions of the string can be exhibited by means of a triangle of forces; and if the successive triangles be drawn to the same scale they can be fitted together so as to constitute a single _force-diagram_, as shown in fig. 6. This diagram consists of a polygon whose successive sides represent the given forces P1, P2, P3 ..., and of a series of lines connecting the vertices with a point O. These latter lines measure the tensions in the successive portions of string. As a special, but very important case, the forces P1, P2, P3 ... may be parallel, e.g. they may be the weights of the several particles. The polygon of forces is then made up of segments of a vertical line. We note that the tensions have now the same horizontal projection (represented by the dotted line in fig. 7). It is further of interest to note that if the weights be all equal, and at equal horizontal intervals, the vertices of the funicular will lie on a parabola whose axis is vertical. To prove this statement, let A, B, C, D ... be successive vertices, and let H, K ... be the middle points of AC, BD ...; then BH, CK ... will be vertical by the hypothesis, and since the geometric sum of [->BA], [->BC] is represented by 2[->BH], the tension in BA: tension in BC: weight at B

as BA: BC: 2BH.

The tensions in the successive portions of the string are therefore proportional to the respective lengths, and the lines BH, CK ... are all equal. Hence AD, BC are parallel and are bisected by the same vertical line; and a parabola with vertical axis can therefore be described through A, B, C, D. The same holds for the four points B, C, D, E and so on; but since a parabola is uniquely determined by the direction of its axis and by three points on the curve, the successive parabolas ABCD, BCDE, CDEF ... must be coincident.

§ 3. _Plane Kinematics of a Rigid Body._--The ideal _rigid body_ is one in which the distance between any two points is invariable. For the present we confine ourselves to the consideration of displacements in two dimensions, so that the body is adequately represented by a thin lamina or plate.

The position of a lamina movable in its own plane is determinate when we know the positions of any two points A, B of it. Since the four co-ordinates (Cartesian or other) of these two points are connected by the relation which expresses the invariability of the length AB, it is plain that virtually three independent elements are required and suffice to specify the position of the lamina. For instance, the lamina may in general be fixed by connecting any three points of it by rigid links to three fixed points in its plane. The three independent elements may be chosen in a variety of ways (e.g. they may be the lengths of the three links in the above example). They may be called (in a generalized sense) the _co-ordinates_ of the lamina. The lamina when perfectly free to move in its own plane is said to have _three degrees of freedom_.

By a theorem due to M. Chasles any displacement whatever of the lamina in its own plane is equivalent to a rotation about some finite or infinitely distant point J. For suppose that in consequence of the displacement a point of the lamina is brought from A to B, whilst the point of the lamina which was originally at B is brought to C. Since AB, BC, are two different positions of the same line in the lamina they are equal, and it is evident that the rotation could have been effected by a rotation about J, the centre of the circle ABC, through an angle AJB. As a special case the three points A, B, C may be in a straight line; J is then at infinity and the displacement is equivalent to a pure _translation_, since every point of the lamina is now displaced parallel to AB through a space equal to AB.

Next, consider any continuous motion of the lamina. The latter may be brought from any one of its positions to a neighbouring one by a rotation about the proper centre. The limiting position J of this centre, when the two positions are taken infinitely close to one another, is called the _instantaneous centre_. If P, P´ be consecutive positions of the same point, and [delta][theta] the corresponding angle of rotation, then ultimately PP´ is at right angles to JP and equal to JP·[delta][theta]. The instantaneous centre will have a certain locus in space, and a certain locus in the lamina. These two loci are called _pole-curves_ or _centrodes_, and are sometimes distinguished as the _space-centrode_ and the _body-centrode_, respectively. In the continuous motion in question the latter curve rolls without slipping on the former (M. Chasles). Consider in fact any series of successive positions 1, 2, 3... of the lamina (fig. 11); and let J12, J23, J34... be the positions in space of the centres of the rotations by which the lamina can be brought from the first position to the second, from the second to the third, and so on. Further, in the position 1, let J12, J´23, J´34 ... be the points of the lamina which have become the successive centres of rotation. The given series of positions will be assumed in succession if we imagine the lamina to rotate first about J12 until J´23 comes into coincidence with J23, then about J23 until J´34 comes into coincidence with J34, and so on. This is equivalent to imagining the polygon J12 J´23 J´34 ..., supposed fixed in the lamina, to roll on the polygon J12 J23 J34 ..., which is supposed fixed in space. By imagining the successive positions to be taken infinitely close to one another we derive the theorem stated. The particular case where both centrodes are circles is specially important in mechanism.

The theory may be illustrated by the case of "three-bar motion." Let ABCD be any quadrilateral formed of jointed links. If, AB being held fixed, the quadrilateral be slightly deformed, it is obvious that the instantaneous centre J will be at the intersection of the straight lines AD, BC, since the displacements of the points D, C are necessarily at right angles to AD, BC, respectively. Hence these displacements are proportional to JD, JC, and therefore to DD´ CC´, where C´D´ is any line drawn parallel to CD, meeting BC, AD in C´, D´, respectively. The determination of the centrodes in three-bar motion is in general complicated, but in one case, that of the "crossed parallelogram" (fig. 13), they assume simple forms. We then have AB = DC and AD = BC, and from the symmetries of the figure it is plain that

AJ + JB = CJ + JD = AD.

Hence the locus of J relative to AB, and the locus relative to CD are equal ellipses of which A, B and C, D are respectively the foci. It may be noticed that the lamina in fig. 9 is not, strictly speaking, fixed, but admits of infinitesimal displacement, whenever the directions of the three links are concurrent (or parallel).

The matter may of course be treated analytically, but we shall only require the formula for infinitely small displacements. If the origin of rectangular axes fixed in the lamina be shifted through a space whose projections on the original directions of the axes are [lambda], [mu], and if the axes are simultaneously turned through an angle [epsilon], the co-ordinates of a point of the lamina, relative to the original axes, are changed from x, y to [lambda] + x cos [epsilon] - y sin [epsilon], [mu] + x sin [epsilon] + y cos [epsilon], or [lambda] + x - y[epsilon], [mu] + x[epsilon] + y, ultimately. Hence the component displacements are ultimately

[delta]x = [lambda] - y[epsilon], [delta]y = [mu] + x[epsilon] (1)

If we equate these to zero we get the co-ordinates of the instantaneous centre.

§ 4. _Plane Statics._--The statics of a rigid body rests on the following two assumptions:--

(i) A force may be supposed to be applied indifferently at any point in its line of action. In other words, a force is of the nature of a "bound" or "localized" vector; it is regarded as resident in a certain line, but has no special reference to any particular point of the line.

(ii) Two forces in intersecting lines may be replaced by a force which is their geometric sum, acting through the intersection. The theory of parallel forces is included as a limiting case. For if O, A, B be any three points, and m, n any scalar quantities, we have in vectors

m · [->OA] + n·[->OB] = (m + n) [->OC], (1)

provided

m · [->CA] + n·[->CB] = 0. (2)

Hence if forces P, Q act in OA, OB, the resultant R will pass through C, provided

m = P/OA, n = Q/OB;

also

R = P·OC/OA + Q·OC/OB, (3)

and

P·AC : Q·CB = OA : OB. (4)

These formulae give a means of constructing the resultant by means of any transversal AB cutting the lines of action. If we now imagine the point O to recede to infinity, the forces P, Q and the resultant R are parallel, and we have

R = P + Q, P·AC = Q·CB. (5)

When P, Q have opposite signs the point C divides AB externally on the side of the greater force. The investigation fails when P + Q = 0, since it leads to an infinitely small resultant acting in an infinitely distant line. A combination of two equal, parallel, but oppositely directed forces cannot in fact be replaced by anything simpler, and must therefore be recognized as an independent entity in statics. It was called by L. Poinsot, who first systematically investigated its properties, a _couple_.

We now restrict ourselves for the present to the systems of forces in one plane. By successive applications of (ii) any such coplanar system can in general be reduced to a _single resultant_ acting in a definite line. As exceptional cases the system may reduce to a couple, or it may be in equilibrium.

The _moment_ of a force about a point O is the product of the force into the perpendicular drawn to its line of action from O, this perpendicular being reckoned positive or negative according as O lies on one side or other of the line of action. If we mark off a segment AB along the line of action so as to represent the force completely, the moment is represented as to magnitude by twice the area of the triangle OAB, and the usual convention as to sign is that the area is to be reckoned positive or negative according as the letters O, A, B, occur in "counter-clockwise" or "clockwise" order.

The sum of the moments of two forces about any point O is equal to the moment of their resultant (P. Varignon, 1687). Let AB, AC (fig. 16) represent the two forces, AD their resultant; we have to prove that the sum of the triangles OAB, OAC is equal to the triangle OAD, regard being had to signs. Since the side OA is common, we have to prove that the sum of the perpendiculars from B and C on OA is equal to the perpendicular from D on OA, these perpendiculars being reckoned positive or negative according as they lie to the right or left of AO. Regarded as a statement concerning the orthogonal projections of the vectors [->AB] and [->AC] (or BD), and of their sum [->AD], on a line perpendicular to AO, this is obvious.

It is now evident that in the process of reduction of a coplanar system no change is made at any stage either in the sum of the projections of the forces on any line or in the sum of their moments about any point. It follows that the single resultant to which the system in general reduces is uniquely determinate, i.e. it acts in a definite line and has a definite magnitude and sense. Again it is necessary and sufficient for equilibrium that the sum of the projections of the forces on each of two perpendicular directions should vanish, and (moreover) that the sum of the moments about some one point should be zero. The fact that three independent conditions must hold for equilibrium is important. The conditions may of course be expressed in different (but equivalent) forms; e.g. the sum of the moments of the forces about each of the three points which are not collinear must be zero.

The particular case of three forces is of interest. If they are not all parallel they must be concurrent, and their vector-sum must be zero. Thus three forces acting perpendicular to the sides of a triangle at the middle points will be in equilibrium provided they are proportional to the respective sides, and act all inwards or all outwards. This result is easily extended to the case of a polygon of any number of sides; it has an important application in hydrostatics.

Again, suppose we have a bar AB resting with its ends on two smooth inclined planes which face each other. Let G be the centre of gravity (§ 11), and let AG = a, GB = b. Let [alpha], [beta] be the inclinations of the planes, and [theta] the angle which the bar makes with the vertical. The position of equilibrium is determined by the consideration that the reactions at A and B, which are by hypothesis normal to the planes, must meet at a point J on the vertical through G. Hence

JG/a = sin ([theta] - [alpha])/sin [alpha], JG/b = sin ([theta] + [beta])/sin [beta],

whence

a cot [alpha] - b cot [beta] cot [theta] = ----------------------------. (6) a + b

If the bar is uniform we have a = b, and

cot [theta] = ½ (cot [alpha] - cot [beta]). (7)

The problem of a rod suspended by strings attached to two points of it is virtually identical, the tensions of the strings taking the place of the reactions of the planes.

Just as a system of forces is in general equivalent to a single force, so a given force can conversely be replaced by combinations of other forces, in various ways. For instance, a given force (and consequently a system of forces) can be replaced in one and only one way by three forces acting in three assigned straight lines, provided these lines be not concurrent or parallel. Thus if the three lines form a triangle ABC, and if the given force F meet BC in H, then F can be resolved into two components acting in HA, BC, respectively. And the force in HA can be resolved into two components acting in BC, CA, respectively. A simple graphical construction is indicated in fig. 19, where the dotted lines are parallel. As an example, any system of forces acting on the lamina in fig. 9 is balanced by three determinate tensions (or thrusts) in the three links, provided the directions of the latter are not concurrent.

If P, Q, R, be any three forces acting along BC, CA, AB, respectively, the line of action of the resultant is determined by the consideration that the sum of the moments about any point on it must vanish. Hence in "trilinear" co-ordinates, with ABC as fundamental triangle, its equation is P[alpha] + Q[beta] + R[gamma] = 0. If P : Q : R = a : b : c, where a, b, c are the lengths of the sides, this becomes the "line at infinity," and the forces reduce to a couple.

The sum of the moments of the two forces of a couple is the same about any point in the plane. Thus in the figure the sum of the moments about O is P·OA - P·OB or P·AB, which is independent of the position of O. This sum is called the _moment of the couple_; it must of course have the proper sign attributed to it. It easily follows that any two couples of the same moment are equivalent, and that any number of couples can be replaced by a single couple whose moment is the sum of their moments. Since a couple is for our purposes sufficiently represented by its moment, it has been proposed to substitute the name _torque_ (or twisting effort), as free from the suggestion of any special pair of forces.

A system of forces represented completely by the sides of a plane polygon taken in order is equivalent to a couple whose moment is represented by twice the area of the polygon; this is proved by taking moments about any point. If the polygon intersects itself, care must be taken to attribute to the different parts of the area their proper signs.

Again, any coplanar system of forces can be replaced by a single force R acting at any assigned point O, together with a couple G. The force R is the geometric sum of the given forces, and the moment (G) of the couple is equal to the sum of the moments of the given forces about O. The value of G will in general vary with the position of O, and will vanish when O lies on the line of action of the single resultant.

The formal analytical reduction of a system of coplanar forces is as follows. Let (x1, y1), (x2, y2), ... be the rectangular co-ordinates of any points A1, A2, ... on the lines of action of the respective forces. The force at A1 may be replaced by its components X1, Y1, parallel to the co-ordinate axes; that at A2 by its components X2, Y2, and so on. Introducing at O two equal and opposite forces ±X1 in Ox, we see that X1 at A1 may be replaced by an equal and parallel force at O together with a couple -y1X1. Similarly the force Y1 at A1 may be replaced by a force Y1 at O together with a couple x1Y1. The forces X1, Y1, at O can thus be transferred to O provided we introduce a couple x1Y1 - y1X1. Treating the remaining forces in the same way we get a force X1 + X2 + ... or [Sigma](X) along Ox, a force Y1 + Y2 + ... or [Sigma](Y) along Oy, and a couple (x1Y1 - y1X1) + (x2Y2 - y2X2) + ... or [Sigma](xY - yX). The three conditions of equilibrium are therefore

[Sigma](X) = 0, [Sigma](Y) = 0, [Sigma](xY - yX) = 0. (8)

If O´ be a point whose co-ordinates are ([xi], [eta]), the moment of the couple when the forces are transferred to O´ as a new origin will be [Sigma]{(x - [xi]) Y - (y - [eta]) X}. This vanishes, i.e. the system reduces to a single resultant through O´, provided

-[xi]·[Sigma](Y) + [eta]·[Sigma](X) + [Sigma](xY - yX) = 0. (9)

If [xi], [eta] be regarded as current co-ordinates, this is the equation of the line of action of the single resultant to which the system is in general reducible.

If the forces are all parallel, making say an angle [theta] with Ox, we may write X1 = P1 cos [theta], Y1 = P1 sin [theta], X2 = P2 cos [theta], Y2 = P2 sin [theta], .... The equation (9) then becomes

{[Sigma](xP) - [xi]·[Sigma](P)} sin [theta] - {[Sigma](yP) - [eta]·[Sigma](P)} cos [theta] = 0. (10)

If the forces P1, P2, ... be turned in the same sense through the same angle about the respective points A1, A2, ... so as to remain parallel, the value of [theta] is alone altered, and the resultant [Sigma](P) passes always through the point

[Sigma](xP) [Sigma](yP) [|x] = -----------, [|y] = -----------, (11) [Sigma](P) [Sigma](P)

which is determined solely by the configuration of the points A1, A2, ... and by the ratios P1: P2: ... of the forces acting at them respectively. This point is called the _centre_ of the given system of parallel forces; it is finite and determinate unless [Sigma](P) = 0. A geometrical proof of this theorem, which is not restricted to a two-dimensional system, is given later (§ 11). It contains the theory of the _centre of gravity_ as ordinarily understood. For if we have an assemblage of particles whose mutual distances are small compared with the dimensions of the earth, the forces of gravity on them constitute a system of sensibly parallel forces, sensibly proportional to the respective masses. If now the assemblage be brought into any other position relative to the earth, without alteration of the mutual distances, this is equivalent to a rotation of the directions of the forces relatively to the assemblage, the ratios of the forces remaining unaltered. Hence there is a certain point, fixed relatively to the assemblage, through which the resultant of gravitational action always passes; this resultant is moreover equal to the sum of the forces on the several particles.

The theorem that any coplanar system of forces can be reduced to a force acting through any assigned point, together with a couple, has an important illustration in the theory of the distribution of shearing stress and bending moment in a horizontal beam, or other structure, subject to vertical extraneous forces. If we consider any vertical section P, the forces exerted across the section by the portion of the structure on one side on the portion on the other may be reduced to a vertical force F at P and a couple M. The force measures the _shearing stress_, and the couple the _bending moment_ at P; we will reckon these quantities positive when the senses are as indicated in the figure.

If the remaining forces acting on the portion of the structure on either side of P are known, then resolving vertically we find F, and taking moments about P we find M. Again if PQ be any segment of the beam which is free from load, Q lying to the right of P, we find

F_P = F_Q, M_P - M_Q = -F·PQ; (12)

hence F is constant between the loads, whilst M decreases as we travel to the right, with a constant gradient -F. If PQ be a short segment containing an isolated load W, we have

F_Q - F_P = -W, M_Q = M_P; (13)

hence F is discontinuous at a concentrated load, diminishing by an amount equal to the load as we pass the loaded point to the right, whilst M is continuous. Accordingly the graph of F for any system of isolated loads will consist of a series of horizontal lines, whilst that of M will be a continuous polygon.

To pass to the case of continuous loads, let x be measured horizontally along the beam to the right. The load on an element [delta]x of the beam may be represented by w[delta]x, where w is in general a function of x. The equations (12) are now replaced by

[delta]F = -w[delta]x, [delta]M = -F[delta]x,

whence _ _ / Q / Q F_Q - F_P = - | w dx, M_Q - M_P = - | F dx. (14) _/P _/P

The latter relation shows that the bending moment varies as the area cut off by the ordinate in the graph of F. In the case of uniform load we have

F = -wx + A, M = ½wx² - Ax + B, (15)

where the arbitrary constants A,B are to be determined by the conditions of the special problem, e.g. the conditions at the ends of the beam. The graph of F is a straight line; that of M is a parabola with vertical axis. In all cases the graphs due to different distributions of load may be superposed. The figure shows the case of a uniform heavy beam supported at its ends.

§ 5. _Graphical Statics._--A graphical method of reducing a plane system of forces was introduced by C. Culmann (1864). It involves the construction of two figures, a _force-diagram_ and a _funicular polygon_. The force-diagram is constructed by placing end to end a series of vectors representing the given forces in magnitude and direction, and joining the vertices of the polygon thus formed to an arbitrary _pole_ O. The funicular or link polygon has its vertices on the lines of action of the given forces, and its sides respectively parallel to the lines drawn from O in the force-diagram; in particular, the two sides meeting in any vertex are respectively parallel to the lines drawn from O to the ends of that side of the force-polygon which represents the corresponding force. The relations will be understood from the annexed diagram, where corresponding lines in the force-diagram (to the right) and the funicular (to the left) are numbered similarly. The sides of the force-polygon may in the first instance be arranged in any order; the force-diagram can then be completed in a doubly infinite number of ways, owing to the arbitrary position of O; and for each force-diagram a simply infinite number of funiculars can be drawn. The two diagrams being supposed constructed, it is seen that each of the given systems of forces can be replaced by two components acting in the sides of the funicular which meet at the corresponding vertex, and that the magnitudes of these components will be given by the corresponding triangle of forces in the force-diagram; thus the force 1 in the figure is equivalent to two forces represented by 01 and 12. When this process of replacement is complete, each terminated side of the funicular is the seat of two forces which neutralize one another, and there remain only two uncompensated forces, viz., those resident in the first and last sides of the funicular. If these sides intersect, the resultant acts through the intersection, and its magnitude and direction are given by the line joining the first and last sides of the force-polygon (see fig. 26, where the resultant of the four given forces is denoted by R). As a special case it may happen that the force-polygon is closed, i.e. its first and last points coincide; the first and last sides of the funicular will then be parallel (unless they coincide), and the two uncompensated forces form a couple. If, however, the first and last sides of the funicular coincide, the two outstanding forces neutralize one another, and we have equilibrium. Hence the necessary and sufficient conditions of equilibrium are that the force-polygon and the funicular should both be closed. This is illustrated by fig. 26 if we imagine the force R, reversed, to be included in the system of given forces.

It is evident that a system of jointed bars having the shape of the funicular polygon would be in equilibrium under the action of the given forces, supposed applied to the joints; moreover any bar in which the stress is of the nature of a tension (as distinguished from a thrust) might be replaced by a string. This is the origin of the names "link-polygon" and "funicular" (cf. § 2).

If funiculars be drawn for two positions O, O´ of the pole in the force-diagram, their corresponding sides will intersect on a straight line parallel to OO´. This is essentially a theorem of projective geometry, but the following statical proof is interesting. Let AB (fig. 27) be any side of the force-polygon, and construct the corresponding portions of the two diagrams, first with O and then with O´ as pole. The force corresponding to AB may be replaced by the two components marked x, y; and a force corresponding to BA may be represented by the two components marked x´, y´. Hence the forces x, y, x´, y´ are in equilibrium. Now x, x´ have a resultant through H, represented in magnitude and direction by OO´, whilst y, y´ have a resultant through K represented in magnitude and direction by O´O. Hence HK must be parallel to OO´. This theorem enables us, when one funicular has been drawn, to construct any other without further reference to the force-diagram.

The complete figures obtained by drawing first the force-diagrams of a system of forces in equilibrium with two distinct poles O, O´, and secondly the corresponding funiculars, have various interesting relations. In the first place, each of these figures may be conceived as an orthogonal projection of a closed plane-faced polyhedron. As regards the former figure this is evident at once; viz. the polyhedron consists of two pyramids with vertices represented by O, O´, and a common base whose perimeter is represented by the force-polygon (only one of these is shown in fig. 28). As regards the funicular diagram, let LM be the line on which the pairs of corresponding sides of the two polygons meet, and through it draw any two planes [omega], [omega]´. Through the vertices A, B, C, ... and A´, B´, C´, ... of the two funiculars draw normals to the plane of the diagram, to meet [omega] and [omega]´ respectively. The points thus obtained are evidently the vertices of a polyhedron with plane faces.

To every line in either of the original figures corresponds of course a parallel line in the other; moreover, it is seen that concurrent lines in either figure correspond to lines forming a closed polygon in the other. Two plane figures so related are called _reciprocal_, since the properties of the first figure in relation to the second are the same as those of the second with respect to the first. A still simpler instance of reciprocal figures is supplied by the case of concurrent forces in equilibrium (fig. 29). The theory of these reciprocal figures was first studied by J. Clerk Maxwell, who showed amongst other things that a reciprocal can always be drawn to any figure which is the orthogonal projection of a plane-faced polyhedron. If in fact we take the pole of each face of such a polyhedron with respect to a paraboloid of revolution, these poles will be the vertices of a second polyhedron whose edges are the "conjugate lines" of those of the former. If we project both polyhedra orthogonally on a plane perpendicular to the axis of the paraboloid, we obtain two figures which are reciprocal, except that corresponding lines are orthogonal instead of parallel. Another proof will be indicated later (§ 8) in connexion with the properties of the linear complex. It is convenient to have a notation which shall put in evidence the reciprocal character. For this purpose we may designate the points in one figure by letters A, B, C, ... and the corresponding polygons in the other figure by the same letters; a line joining two points A, B in one figure will then correspond to the side common to the two polygons A, B in the other. This notation was employed by R. H. Bow in connexion with the theory of frames (§ 6, and see also APPLIED MECHANICS below) where reciprocal diagrams are frequently of use (cf. DIAGRAM).

When the given forces are all parallel, the force-polygon consists of a series of segments of a straight line. This case has important practical applications; for instance we may use the method to find the pressures on the supports of a beam loaded in any given manner. Thus if AB, BC, CD represent the given loads, in the force-diagram, we construct the sides corresponding to OA, OB, OC, OD in the funicular; we then draw the _closing line_ of the funicular polygon, and a parallel OE to it in the force diagram. The segments DE, EA then represent the upward pressures of the two supports on the beam, which pressures together with the given loads constitute a system of forces in equilibrium. The pressures of the beam on the supports are of course represented by ED, AE. The two diagrams are portions of reciprocal figures, so that Bow's notation is applicable.

A graphical method can also be applied to find the moment of a force, or of a system of forces, about any assigned point P. Let F be a force represented by AB in the force-diagram. Draw a parallel through P to meet the sides of the funicular which correspond to OA, OB in the points H, K. If R be the intersection of these sides, the triangles OAB, RHK are similar, and if the perpendiculars OM, RN be drawn we have

HK·OM = AB·RN = F·RN,

which is the moment of F about P. If the given forces are all parallel (say vertical) OM is the same for all, and the moments of the several forces about P are represented on a certain scale by the lengths intercepted by the successive pairs of sides on the vertical through P. Moreover, the moments are compounded by adding (geometrically) the corresponding lengths HK. Hence if a system of vertical forces be in equilibrium, so that the funicular polygon is closed, the length which this polygon intercepts on the vertical through any point P gives the sum of the moments about P of all the forces on one side of this vertical. For instance, in the case of a beam in equilibrium under any given loads and the reactions at the supports, we get a graphical representation of the distribution of bending moment over the beam. The construction in fig. 30 can easily be adjusted so that the closing line shall be horizontal; and the figure then becomes identical with the bending-moment diagram of § 4. If we wish to study the effects of a movable load, or system of loads, in different positions on the beam, it is only necessary to shift the lines of action of the pressures of the supports relatively to the funicular, keeping them at the same, distance apart; the only change is then in the position of the closing line of the funicular. It may be remarked that since this line joins homologous points of two "similar" rows it will envelope a parabola.

The "centre" (§ 4) of a system of parallel forces of given magnitudes, acting at given points, is easily determined graphically. We have only to construct the line of action of the resultant for each of two arbitrary directions of the forces; the intersection of the two lines gives the point required. The construction is neatest if the two arbitrary directions are taken at right angles to one another.

§ 6. _Theory of Frames._--A _frame_ is a structure made up of pieces, or _members_, each of which has two _joints_ connecting it with other members. In a two-dimensional frame, each joint may be conceived as consisting of a small cylindrical pin fitting accurately and smoothly into holes drilled through the members which it connects. This supposition is a somewhat ideal one, and is often only roughly approximated to in practice. We shall suppose, in the first instance, that extraneous forces act on the frame at the joints only, i.e. on the pins.

On this assumption, the reactions on any member at its two joints must be equal and opposite. This combination of equal and opposite forces is called the _stress_ in the member; it may be a _tension_ or a _thrust_. For diagrammatic purposes each member is sufficiently represented by a straight line terminating at the two joints; these lines will be referred to as the _bars_ of the frame.

In structural applications a frame must be _stiff_, or _rigid_, i.e. it must be incapable of deformation without alteration of length in at least one of its bars. It is said to be _just rigid_ if it ceases to be rigid when any one of its bars is removed. A frame which has more bars than are essential for rigidity may be called _over-rigid_; such a frame is in general self-stressed, i.e. it is in a state of stress independently of the action of extraneous forces. A plane frame of n joints which is just rigid (as regards deformation in its own plane) has 2n - 3 bars, for if one bar be held fixed the 2(n - 2) co-ordinates of the remaining n - 2 joints must just be determined by the lengths of the remaining bars. The total number of bars is therefore 2(n - 2) + 1. When a plane frame which is just rigid is subject to a given system of equilibrating extraneous forces (in its own plane) acting on the joints, the stresses in the bars are in general uniquely determinate. For the conditions of equilibrium of the forces on each pin furnish 2n equations, viz. two for each point, which are linear in respect of the stresses and the extraneous forces. This system of equations must involve the three conditions of equilibrium of the extraneous forces which are already identically satisfied, by hypothesis; there remain therefore 2n - 3 independent relations to determine the 2n - 3 unknown stresses. A frame of n joints and 2n - 3 bars may of course fail to be rigid owing to some parts being over-stiff whilst others are deformable; in such a case it will be found that the statical equations, apart from the three identical relations imposed by the equilibrium of the extraneous forces, are not all independent but are equivalent to less than 2n - 3 relations. Another exceptional case, known as the _critical case_, will be noticed later (§ 9).

A plane frame which can be built up from a single bar by successive steps, at each of which a new joint is introduced by two new bars meeting there, is called a _simple_ frame; it is obviously just rigid. The stresses produced by extraneous forces in a simple frame can be found by considering the equilibrium of the various joints in a proper succession; and if the graphical method be employed the various polygons of force can be combined into a single force-diagram. This procedure was introduced by W. J. M. Rankine and J. Clerk Maxwell (1864). It may be noticed that if we take an arbitrary pole in the force-diagram, and draw a corresponding funicular in the skeleton diagram which represents the frame together with the lines of action of the extraneous forces, we obtain two complete reciprocal figures, in Maxwell's sense. It is accordingly convenient to use Bow's notation (§ 5), and to distinguish the several compartments of the frame-diagram by letters. See fig. 33, where the successive triangles in the diagram of forces may be constructed in the order XYZ, ZXA, AZB. The class of "simple" frames includes many of the frameworks used in the construction of roofs, lattice girders and suspension bridges; a number of examples will be found in the article BRIDGES. By examining the senses in which the respective forces act at each joint we can ascertain which members are in tension and which are in thrust; in fig. 33 this is indicated by the directions of the arrowheads.

When a frame, though just rigid, is not "simple" in the above sense, the preceding method must be replaced, or supplemented, by one or other of various artifices. In some cases the _method of sections_ is sufficient for the purpose. If an ideal section be drawn across the frame, the extraneous forces on either side must be in equilibrium with the forces in the bars cut across; and if the section can be drawn so as to cut only three bars, the forces in these can be found, since the problem reduces to that of resolving a given force into three components acting in three given lines (§ 4). The "critical case" where the directions of the three bars are concurrent is of course excluded. Another method, always available, will be explained under "Work" (§ 9).

When extraneous forces act on the bars themselves the stress in each bar no longer consists of a simple longitudinal tension or thrust. To find the reactions at the joints we may proceed as follows. Each extraneous force W acting on a bar may be replaced (in an infinite number of ways) by two components P, Q in lines through the centres of the pins at the extremities. In practice the forces W are usually vertical, and the components P, Q are then conveniently taken to be vertical also. We first alter the problem by transferring the forces P, Q to the pins. The stresses in the bars, in the problem as thus modified, may be supposed found by the preceding methods; it remains to infer from the results thus obtained the reactions in the original form of the problem. To find the pressure exerted by a bar AB on the pin A we compound with the force in AB given by the diagram a force equal to P. Conversely, to find the pressure of the pin A on the bar AB we must compound with the force given by the diagram a force equal and opposite to P. This question arises in practice in the theory of "three-jointed" structures; for the purpose in hand such a structure is sufficiently represented by two bars AB, BC. The right-hand figure represents a portion of the force-diagram; in particular [->ZX] represents the pressure of AB on B in the modified problem where the loads W1 and W2 on the two bars are replaced by loads P1, Q1, and P2, Q2 respectively, acting on the pins. Compounding with this [->XV], which represents Q1, we get the actual pressure [->ZV] exerted by AB on B. The directions and magnitudes of the reactions at A and C are then easily ascertained. On account of its practical importance several other graphical solutions of this problem have been devised.

§ 7. _Three-dimensional Kinematics of a Rigid Body._--The position of a rigid body is determined when we know the positions of three points A, B, C of it which are not collinear, for the position of any other point P is then determined by the three distances PA, PB, PC. The nine co-ordinates (Cartesian or other) of A, B, C are subject to the three relations which express the invariability of the distances BC, CA, AB, and are therefore equivalent to six independent quantities. Hence a rigid body not constrained in any way is said to have six degrees of freedom. Conversely, any six geometrical relations restrict the body in general to one or other of a series of definite positions, none of which can be departed from without violating the conditions in question. For instance, the position of a theodolite is fixed by the fact that its rounded feet rest in contact with six given plane surfaces. Again, a rigid three-dimensional frame can be rigidly fixed relatively to the earth by means of six links.

The six independent quantities, or "co-ordinates," which serve to specify the position of a rigid body in space may of course be chosen in an endless variety of ways. We may, for instance, employ the three Cartesian co-ordinates of a particular point O of the body, and three angular co-ordinates which express the orientation of the body with respect to O. Thus in fig. 36, if OA, OB, OC be three mutually perpendicular lines in the solid, we may denote by [theta] the angle which OC makes with a fixed direction OZ, by [psi] the azimuth of the plane ZOC measured from some fixed plane through OZ, and by [phi] the inclination of the plane COA to the plane ZOC. In fig. 36 these various lines and planes are represented by their intersections with a unit sphere having O as centre. This very useful, although unsymmetrical, system of angular co-ordinates was introduced by L. Euler. It is exemplified in "Cardan's suspension," as used in connexion with a compass-bowl or a gyroscope. Thus in the gyroscope the "flywheel" (represented by the globe in fig. 37) can turn about a diameter OC of a ring which is itself free to turn about a diametral axis OX at right angles to the former; this axis is carried by a second ring which is free to turn about a fixed diameter OZ, which is at right angles to OX.

We proceed to sketch the theory of the finite displacements of a rigid body. It was shown by Euler (1776) that any displacement in which one point O of the body is fixed is equivalent to a pure _rotation_ about some axis through O. Imagine two spheres of equal radius with O as their common centre, one fixed in the body and moving with it, the other fixed in space. In any displacement about O as a fixed point, the former sphere slides over the latter, as in a "ball-and-socket" joint. Suppose that as the result of the displacement a point of the moving sphere is brought from A to B, whilst the point which was at B is brought to C (cf. fig. 10). Let J be the pole of the circle ABC (usually a "small circle" of the fixed sphere), and join JA, JB, JC, AB, BC by great-circle arcs. The spherical isosceles triangles AJB, BJC are congruent, and we see that AB can be brought into the position BC by a rotation about the axis OJ through an angle AJB.

It is convenient to distinguish the two senses in which rotation may take place about an axis OA by opposite signs. We shall reckon a rotation as positive when it is related to the direction from O to A as the direction of rotation is related to that of translation in a right-handed screw. Thus a negative rotation about OA may be regarded as a positive rotation about OA´, the prolongation of AO. Now suppose that a body receives first a positive rotation [alpha] about OA, and secondly a positive rotation [beta] about OB; and let A, B be the intersections of these axes with a sphere described about O as centre. If we construct the spherical triangles ABC, ABC´ (fig. 38), having in each case the angles at A and B equal to ½[alpha] and ½[beta] respectively, it is evident that the first rotation will bring a point from C to C´ and that the second will bring it back to C; the result is therefore equivalent to a rotation about OC. We note also that if the given rotations had been effected in the inverse order, the axis of the resultant rotation would have been OC´, so that finite rotations do not obey the "commutative law." To find the angle of the equivalent rotation, in the actual case, suppose that the second rotation (about OB) brings a point from A to A´. The spherical triangles ABC, A´BC (fig. 39) are "symmetrically equal," and the angle of the resultant rotation, viz. ACA´, is 2[pi] - 2C. This is equivalent to a negative rotation 2C about OC, whence the theorem that the effect of three successive positive rotations 2A, 2B, 2C about OA, OB, OC, respectively, is to leave the body in its original position, provided the circuit ABC is left-handed as seen from O. This theorem is due to O. Rodrigues (1840). The composition of finite rotations about parallel axes is a particular case of the preceding; the radius of the sphere is now infinite, and the triangles are plane.

In any continuous motion of a solid about a fixed point O, the limiting position of the axis of the rotation by which the body can be brought from any one of its positions to a consecutive one is called the _instantaneous axis_. This axis traces out a certain cone in the body, and a certain cone in space, and the continuous motion in question may be represented as consisting in a rolling of the former cone on the latter. The proof is similar to that of the corresponding theorem of plane kinematics (§ 3).

It follows from Euler's theorem that the most general displacement of a rigid body may be effected by a pure translation which brings any one point of it to its final position O, followed by a pure rotation about some axis through O. Those planes in the body which are perpendicular to this axis obviously remain parallel to their original positions. Hence, if [sigma], [sigma]´ denote the initial and final positions of any figure in one of these planes, the displacement could evidently have been effected by (1) a translation perpendicular to the planes in question, bringing [sigma] into some position [sigma]´´ in the plane of [sigma]´, and (2) a rotation about a normal to the planes, bringing [sigma]´´ into coincidence with [sigma] (§ 3). In other words, the most general displacement is equivalent to a translation parallel to a certain axis combined with a rotation about that axis; i.e. it may be described as a _twist_ about a certain _screw_. In particular cases, of course, the translation, or the rotation, may vanish.

The preceding theorem, which is due to Michel Chasles (1830), may be proved in various other interesting ways. Thus if a point of the body be displaced from A to B, whilst the point which was at B is displaced to C, and that which was at C to D, the four points A, B, C, D lie on a helix whose axis is the common perpendicular to the bisectors of the angles ABC, BCD. This is the axis of the required screw; the amount of the translation is measured by the projection of AB or BC or CD on the axis; and the angle of rotation is given by the inclination of the aforesaid bisectors. This construction was given by M. W. Crofton. Again, H. Wiener and W. Burnside have employed the _half-turn_ (i.e. a rotation through two right angles) as the fundamental operation. This has the advantage that it is completely specified by the axis of the rotation, the sense being immaterial. Successive half-turns about parallel axes a, b are equivalent to a translation measured by double the distance between these axes in the direction from a to b. Successive half-turns about intersecting axes a, b are equivalent to a rotation about the common perpendicular to a, b at their intersection, of amount equal to twice the acute angle between them, in the direction from a to b. Successive half-turns about two skew axes a, b are equivalent to a twist about a screw whose axis is the common perpendicular to a, b, the translation being double the shortest distance, and the angle of rotation being twice the acute angle between a, b, in the direction from a to b. It is easily shown that any displacement whatever is equivalent to two half-turns and therefore to a screw.

In mechanics we are specially concerned with the theory of infinitesimal displacements. This is included in the preceding, but it is simpler in that the various operations are commutative. An infinitely small rotation about any axis is conveniently represented geometrically by a length AB measures along the axis and proportional to the angle of rotation, with the convention that the direction from A to B shall be related to the rotation as is the direction of translation to that of rotation in a right-handed screw. The consequent displacement of any point P will then be at right angles to the plane PAB, its amount will be represented by double the area of the triangle PAB, and its sense will depend on the cyclical order of the letters P, A, B. If AB, AC represent infinitesimal rotations about intersecting axes, the consequent displacement of any point O in the plane BAC will be at right angles to this plane, and will be represented by twice the sum of the areas OAB, OAC, taken with proper signs. It follows by analogy with the theory of moments (§ 4) that the resultant rotation will be represented by AD, the vector-sum of AB, AC (see fig. 16). It is easily inferred as a limiting case, or proved directly, that two infinitesimal rotations [alpha], [beta] about parallel axes are equivalent to a rotation [alpha] + [beta] about a parallel axis in the same plane with the two former, and dividing a common perpendicular AB in a point C so that AC/CB = [beta]/[alpha]. If the rotations are equal and opposite, so that [alpha] + [beta] = 0, the point C is at infinity, and the effect is a translation perpendicular to the plane of the two given axes, of amount [alpha]·AB. It thus appears that an infinitesimal rotation is of the nature of a "localized vector," and is subject in all respects to the same mathematical laws as a force, conceived as acting on a rigid body. Moreover, that an infinitesimal translation is analogous to a couple and follows the same laws. These results are due to Poinsot.

The analytical treatment of small displacements is as follows. We first suppose that one point O of the body is fixed, and take this as the origin of a "right-handed" system of rectangular co-ordinates; i.e. the positive directions of the axes are assumed to be so arranged that a positive rotation of 90° about Ox would bring Oy into the position of Oz, and so on. The displacement will consist of an infinitesimal rotation [epsilon] about some axis through O, whose direction-cosines are, say, l, m, n. From the equivalence of a small rotation to a localized vector it follows that the rotation [epsilon] will be equivalent to rotations [xi], [eta], [zeta] about Ox, Oy, Oz, respectively, provided

[xi] = l[epsilon], [eta] = m[epsilon], [zeta] = n[epsilon], (1)

and we note that

[xi]² + [eta]² + [zeta]² = [epsilon]². (2)

Thus in the case of fig. 36 it may be required to connect the infinitesimal rotations [xi], [eta], [zeta] about OA, OB, OC with the variations of the angular co-ordinates [theta], [psi], [phi]. The displacement of the point C of the body is made up of [delta][theta] tangential to the meridian ZC and sin [theta] [delta][psi] perpendicular to the plane of this meridian. Hence, resolving along the tangents to the arcs BC, CA, respectively, we have

[xi] = [delta][theta] sin [phi] - sin [theta] [delta][psi] cos [phi], [eta] = [delta][theta] cos [phi] + sin [theta] [delta][psi] sin [phi]. (3)

Again, consider the point of the solid which was initially at A´ in the figure. This is displaced relatively to A´ through a space [delta][psi] perpendicular to the plane of the meridian, whilst A´ itself is displaced through a space cos [theta] [delta][psi] in the same direction. Hence

[zeta] = [delta][phi] + cos [theta] [delta][psi]. (4)

To find the component displacements of a point P of the body, whose co-ordinates are x, y, z, we draw PL normal to the plane yOz, and LH, LK perpendicular to Oy, Oz, respectively. The displacement of P parallel to Ox is the same as that of L, which is made up of [eta]z and -[zeta]y. In this way we obtain the formulae

[delta]x = [eta]z - [zeta]y, [delta]y = [zeta]x - [xi]z, [delta]z = [xi]y - [eta]x. (5)

The most general case is derived from this by adding the component displacements [lambda], [mu], [nu] (say) of the point which was at O; thus

[delta]x = [lambda] + [eta]z - [zeta]y, \ [delta]y = [mu] + [zeta]x - [xi]z, > (6) [delta]z = [nu] + [xi]y - [eta]x. /

The displacement is thus expressed in terms of the six independent quantities [xi], [eta], [zeta], [lambda], [mu], [nu]. The points whose displacements are in the direction of the resultant axis of rotation are determined by [delta]x:[delta]y:[delta]z = [xi]:[eta]:[zeta], or

([lambda] + [eta]z - [zeta]y)/([xi] = [mu] + [zeta]x - [xi]z)/[eta] = ([nu] + [xi]y - [eta]x)/[zeta]. (7)

These are the equations of a straight line, and the displacement is in fact equivalent to a twist about a screw having this line as axis. The translation parallel to this axis is

l[delta]x + m[delta]y + n[delta]z = ([lambda][xi] + [mu][eta] + [nu][zeta])/[epsilon]. (8)

The linear magnitude which measures the ratio of translation to rotation in a screw is called the _pitch_. In the present case the pitch is

([lambda][xi] + [mu][eta] + [nu][zeta])/([xi]² + [eta]² + [zeta]²). (9)

Since [xi]² + [eta]² + [zeta]², or [epsilon]², is necessarily an absolute invariant for all transformations of the (rectangular) co-ordinate axes, we infer that [lambda][xi] + [mu][eta] + [nu][zeta] is also an absolute invariant. When the latter invariant, but not the former, vanishes, the displacement is equivalent to a pure rotation.

If the small displacements of a rigid body be subject to one constraint, e.g. if a point of the body be restricted to lie on a given surface, the mathematical expression of this fact leads to a homogeneous linear equation between the infinitesimals [xi], [eta], [zeta], [lambda], [mu], [nu], say

A[xi] + B[eta] + C[zeta] + F[lambda] + G[mu] + H[nu] = 0. (10)

The quantities [xi], [eta], [zeta], [lambda], [mu], [nu] are no longer independent, and the body has now only five degrees of freedom. Every additional constraint introduces an additional equation of the type (10) and reduces the number of degrees of freedom by one. In Sir R. S. Ball's _Theory of Screws_ an analysis is made of the possible displacements of a body which has respectively two, three, four, five degrees of freedom. We will briefly notice the case of two degrees, which involves an interesting generalization of the method (already explained) of compounding rotations about intersecting axes. We assume that the body receives arbitrary twists about two given screws, and it is required to determine the character of the resultant displacement. We examine first the case where the axes of the two screws are at right angles and intersect. We take these as axes of x and y; then if [xi], [eta] be the component rotations about them, we have

[lambda] = h[xi], [mu] = k[eta], [nu] = 0, (11)

where h, k, are the pitches of the two given screws. The equations (7) of the axis of the resultant screw then reduce to

x/[xi] = y/[eta], z([xi]² + [eta]²) = (k - h)[xi][eta]. (12)

Hence, whatever the ratio [xi] : [eta], the axis of the resultant screw lies on the conoidal surface

z(x² + y²) = cxy, (13)

where c = ½(k - h). The co-ordinates of any point on (13) may be written

x = r cos [theta], y = r sin [theta], z = c sin 2[theta]; (14)

hence if we imagine a curve of sines to be traced on a circular cylinder so that the circumference just includes two complete undulations, a straight line cutting the axis of the cylinder at right angles and meeting this curve will generate the surface. This is called a _cylindroid_. Again, the pitch of the resultant screw is

p = ([lambda][xi] + [mu][eta])/([xi]² + [eta]²) = h cos² [theta] + k sin² [theta]. (15)

The distribution of pitch among the various screws has therefore a simple relation to the _pitch-conic_

hx² + ky² = const; (16)

viz. the pitch of any screw varies inversely as the square of that diameter of the conic which is parallel to its axis. It is to be noticed that the parameter c of the cylindroid is unaltered if the two pitches h, k be increased by equal amounts; the only change is that all the pitches are increased by the same amount. It remains to show that a system of screws of the above type can be constructed so as to contain any two given screws whatever. In the first place, a cylindroid can be constructed so as to have its axis coincident with the common perpendicular to the axes of the two given screws and to satisfy three other conditions, for the position of the centre, the parameter, and the orientation about the axis are still at our disposal. Hence we can adjust these so that the surface shall contain the axes of the two given screws as generators, and that the difference of the corresponding pitches shall have the proper value. It follows that when a body has two degrees of freedom it can twist about any one of a singly infinite system of screws whose axes lie on a certain cylindroid. In particular cases the cylindroid may degenerate into a plane, the pitches being then all equal.

§ 8. _Three-dimensional Statics._--A system of parallel forces can be combined two and two until they are replaced by a single resultant equal to their sum, acting in a certain line. As special cases, the system may reduce to a couple, or it may be in equilibrium.

In general, however, a three-dimensional system of forces cannot be replaced by a single resultant force. But it may be reduced to simpler elements in a variety of ways. For example, it may be reduced to two forces in perpendicular skew lines. For consider any plane, and let each force, at its intersection with the plane, be resolved into two components, one (P) normal to the plane, the other (Q) in the plane. The assemblage of parallel forces P can be replaced in general by a single force, and the coplanar system of forces Q by another single force.

If the plane in question be chosen perpendicular to the direction of the vector-sum of the given forces, the vector-sum of the components Q is zero, and these components are therefore equivalent to a couple (§ 4). Hence any three-dimensional system can be reduced to a single force R acting in a certain line, together with a couple G in a plane perpendicular to the line. This theorem was first given by L. Poinsot, and the line of action of R was called by him the _central axis_ of the system. The combination of a force and a couple in a perpendicular plane is termed by Sir R. S. Ball a _wrench_. Its type, as distinguished from its absolute magnitude, may be specified by a screw whose axis is the line of action of R, and whose pitch is the ratio G/R.

The case of two forces may be specially noticed. Let AB be the shortest distance between the lines of action, and let AA´, BB´ (fig. 42) represent the forces. Let [alpha], [beta] be the angles which AA´, BB´ make with the direction of the vector-sum, on opposite sides. Divide AB in O, so that

AA´·cos [alpha]·AO = BB´·cos [beta]·OB, (1)

and draw OC parallel to the vector-sum. Resolving AA´, BB´ each into two components parallel and perpendicular to OC, we see that the former components have a single resultant in OC, of amount

R = AA´ cos [alpha] + BB´ cos [beta], (2)

whilst the latter components form a couple of moment

G = AA´·AB·sin [alpha] = BB´·AB·sin [beta]. (3)

Conversely it is seen that any wrench can be replaced in an infinite number of ways by two forces, and that the line of action of one of these may be chosen quite arbitrarily. Also, we find from (2) and (3) that

G·R = AA´·BB´·AB·sin ([alpha] + [beta]). (4)

The right-hand expression is six times the volume of the tetrahedron of which the lines AA´, BB´ representing the forces are opposite edges; and we infer that, in whatever way the wrench be resolved into two forces, the volume of this tetrahedron is invariable.

To define the _moment_ of a force _about an axis_ HK, we project the force orthogonally on a plane perpendicular to HK and take the moment of the projection about the intersection of HK with the plane (see § 4). Some convention as to sign is necessary; we shall reckon the moment to be positive when the tendency of the force is right-handed as regards the direction from H to K. Since two concurrent forces and their resultant obviously project into two concurrent forces and their resultant, we see that the sum of the moments of two concurrent forces about any axis HK is equal to the moment of their resultant. Parallel forces may be included in this statement as a limiting case. Hence, in whatever way one system of forces is by successive steps replaced by another, no change is made in the sum of the moments about any assigned axis. By means of this theorem we can show that the previous reduction of any system to a wrench is unique.

From the analogy of couples to translations which was pointed out in § 7, we may infer that a couple is sufficiently represented by a "free" (or non-localized) vector perpendicular to its plane. The length of the vector must be proportional to the moment of the couple, and its sense must be such that the sum of the moments of the two forces of the couple about it is positive. In particular, we infer that couples of the same moment in parallel planes are equivalent; and that couples in any two planes may be compounded by geometrical addition of the corresponding vectors. Independent statical proofs are of course easily given. Thus, let the plane of the paper be perpendicular to the planes of two couples, and therefore perpendicular to the line of intersection of these planes. By § 4, each couple can be replaced by two forces ± P (fig. 43) perpendicular to the plane of the paper, and so that one force of each couple is in the line of intersection (B); the arms (AB, BC) will then be proportional to the respective moments. The two forces at B will cancel, and we are left with a couple of moment P · AC in the plane AC. If we draw three vectors to represent these three couples, they will be perpendicular and proportional to the respective sides of the triangle ABC; hence the third vector is the geometric sum of the other two. Since, in this proof the magnitude of P is arbitrary, It follows incidentally that couples of the same moment in parallel planes, e.g. planes parallel to AC, are equivalent.

Hence a couple of moment G, whose axis has the direction (l, m, n) relative to a right-handed system of rectangular axes, is equivalent to three couples lG, mG, nG in the co-ordinate planes. The analytical reduction of a three-dimensional system can now be conducted as follows. Let (x1, y1, z1) be the co-ordinates of a point P1 on the line of action of one of the forces, whose components are (say) X1, Y1, Z1. Draw P1H normal to the plane zOx, and HK perpendicular to Oz. In KH introduce two equal and opposite forces ± X1. The force X1 at P1 with -X1 in KH forms a couple about Oz, of moment -y1X1. Next, introduce along Ox two equal and opposite forces ±X1. The force X1 in KH with -X1 in Ox forms a couple about Oy, of moment z1X1. Hence the force X1 can be transferred from P1 to O, provided we introduce couples of moments z1X1 about Oy and -y1X1, about Oz. Dealing in the same way with the forces Y1, Z1 at P1, we find that all three components of the force at P1 can be transferred to O, provided we introduce three couples L1, M1, N1 about Ox, Oy, Oz respectively, viz.

L1 = y1Z1 - z1Y1, M1 = z1X1 - x1Z1, N1 = x1Y1 - y1X1. (5)

It is seen that L1, M1, N1 are the moments of the original force at P1 about the co-ordinate axes. Summing up for all the forces of the given system, we obtain a force R at O, whose components are

X = [Sigma](X_r), Y = [Sigma](Y_r), Z = [Sigma](Z_r), (6)

and a couple G whose components are

L = [Sigma](L_r), M = [Sigma](M_r), N = [Sigma](N_r), (7)

where r= 1, 2, 3 ... Since R² = X² + Y² + Z², G² = L² + M² + N², it is necessary and sufficient for equilibrium that the six quantities X, Y, Z, L, M, N, should all vanish. In words: the sum of the projections of the forces on each of the co-ordinate axes must vanish; and, the sum of the moments of the forces about each of these axes must vanish.

If any other point O´, whose co-ordinates are x, y, z, be chosen in place of O, as the point to which the forces are transferred, we have to write x1 - x, y1 - y, z1 - z for x1, y1, z1, and so on, in the preceding process. The components of the resultant force R are unaltered, but the new components of couple are found to be

L´ = L - yZ + zY, \ M´ = M - zX + xZ, > (8) N´ = N - xY + yX. /

By properly choosing O´ we can make the plane of the couple perpendicular to the resultant force. The conditions for this are L´ : M´ : N´ = X : Y : Z, or

L - yZ + zY M - zX + xZ N - xY + yX ----------- = ----------- = ----------- (9) X Y Z

These are the equations of the central axis. Since the moment of the resultant couple is now

X Y Z LX + MY + NZ G´ = --- L´ + --- M´ + --- N´ = ------------, (10) R R R R

the pitch of the equivalent wrench is

(LX + MY + NZ)/(X² + Y² + Z²).

It appears that X² + Y² + Z² and LX + MY + NZ are absolute invariants (cf. § 7). When the latter invariant, but not the former, vanishes, the system reduces to a single force.

The analogy between the mathematical relations of infinitely small displacements on the one hand and those of force-systems on the other enables us immediately to convert any theorem in the one subject into a theorem in the other. For example, we can assert without further proof that any infinitely small displacement may be resolved into two rotations, and that the axis of one of these can be chosen arbitrarily. Again, that wrenches of arbitrary amounts about two given screws compound into a wrench the locus of whose axis is a cylindroid.

The mathematical properties of a twist or of a wrench have been the subject of many remarkable investigations, which are, however, of secondary importance from a physical point of view. In the "Null-System" of A. F. Möbius (1790-1868), a line such that the moment of a given wrench about it is zero is called a _null-line_. The triply infinite system of null-lines form what is called in line-geometry a "complex." As regards the configuration of this complex, consider a line whose shortest distance from the central axis is r, and whose inclination to the central axis is [theta]. The moment of the resultant force R of the wrench about this line is - Rr sin [theta], and that of the couple G is G cos [theta]. Hence the line will be a null-line provided

tan [theta] = k/r, (11)

where k is the pitch of the wrench. The null-lines which are at a given distance r from a point O of the central axis will therefore form one system of generators of a hyperboloid of revolution; and by varying r we get a series of such hyperboloids with a common centre and axis. By moving O along the central axis we obtain the whole complex of null-lines. It appears also from (11) that the null-lines whose distance from the central axis is r are tangent lines to a system of helices of slope tan^-1 (r/k); and it is to be noticed that these helices are left-handed if the given wrench is right-handed, and vice versa.

Since the given wrench can be replaced by a force acting through any assigned point P, and a couple, the locus of the null-lines through P is a plane, viz. a plane perpendicular to the vector which represents the couple. The complex is therefore of the type called "linear" (in relation to the degree of this locus). The plane in question is called the _null-plane_ of P. If the null-plane of P pass through Q, the null-plane of Q will pass through P, since PQ is a null-line. Again, any plane [omega] is the locus of a system of null-lines meeting in a point, called the _null-point_ of [omega]. If a plane revolve about a fixed straight line p in it, its null-point describes another straight line p´, which is called the _conjugate line_ of p. We have seen that the wrench may be replaced by two forces, one of which may act in any arbitrary line p. It is now evident that the second force must act in the conjugate line p´, since every line meeting p, p´ is a null-line. Again, since the shortest distance between any two conjugate lines cuts the central axis at right angles, the orthogonal projections of two conjugate lines on a plane perpendicular to the central axis will be parallel (fig. 42). This property was employed by L. Cremona to prove the existence under certain conditions of "reciprocal figures" in a plane (§ 5). If we take any polyhedron with plane faces, the null-planes of its vertices with respect to a given wrench will form another polyhedron, and the edges of the latter will be conjugate (in the above sense) to those of the former. Projecting orthogonally on a plane perpendicular to the central axis we obtain two reciprocal figures.

In the analogous theory of infinitely small displacements of a solid, a "null-line" is a line such that the lengthwise displacement of any point on it is zero.

Since a wrench is defined by six independent quantities, it can in general be replaced by any system of forces which involves six adjustable elements. For instance, it can in general be replaced by six forces acting in six given lines, e.g. in the six edges of a given tetrahedron. An exception to the general statement occurs when the six lines are such that they are possible lines of action of a system of six forces in equilibrium; they are then said to be _in involution_. The theory of forces in involution has been studied by A. Cayley, J. J. Sylvester and others. We have seen that a rigid structure may in general be rigidly connected with the earth by six links, and it now appears that any system of forces acting on the structure can in general be balanced by six determinate forces exerted by the links. If, however, the links are in involution, these forces become infinite or indeterminate. There is a corresponding kinematic peculiarity, in that the connexion is now not strictly rigid, an infinitely small relative displacement being possible. See § 9.

When parallel forces of given magnitudes act at given points, the resultant acts through a definite point, or _centre of parallel forces_, which is independent of the special direction of the forces. If P_r be the force at (x_r, y_r, z_r), acting in the direction (l, m, n), the formulae (6) and (7) reduce to

X = [Sigma](P).l, Y = [Sigma](P).m, Z = [Sigma](P).n, (12)

and

L = [Sigma](P)·(n[|y] - m[|z]), M = [Sigma](P)·(l[|z] - n[|x]), N = [Sigma](P)·(m[|x] - l[|y]), (13)

provided

[Sigma](Px) [Sigma](Py) [Sigma](Pz) [|x] = -----------, [|y] = -----------, [|z] = -----------. (14) [Sigma](P) [Sigma](P) [Sigma](P)

These are the same as if we had a single force [Sigma](P) acting at the point ([|x], [|y], [|z]), which is the same for all directions (l, m, n). We can hence derive the theory of the centre of gravity, as in § 4. An exceptional case occurs when [Sigma](P) = 0.

If we imagine a rigid body to be acted on at given points by forces of given magnitudes in directions (not all parallel) which are fixed in space, then as the body is turned about the resultant wrench will assume different configurations in the body, and will in certain positions reduce to a single force. The investigation of such questions forms the subject of "Astatics," which has been cultivated by Möbius, Minding, G. Darboux and others. As it has no physical bearing it is passed over here.

§ 9. _Work._--The _work_ done by a force acting on a particle, in any infinitely small displacement, is defined as the product of the force into the orthogonal projection of the displacement on the direction of the force; i.e. it is equal to F·[delta]s cos [theta], where F is the force, [delta]s the displacement, and [theta] is the angle between the directions of F and [delta]s. In the language of vector analysis (q.v.) it is the "scalar product" of the vector representing the force and the displacement. In the same way, the work done by a force acting on a rigid body in any infinitely small displacement of the body is the scalar product of the force into the displacement of any point on the line of action. This product is the same whatever point on the line of action be taken, since the lengthwise components of the displacements of any two points A, B on a line AB are equal, to the first order of small quantities. To see this, let A´, B´ be the displaced positions of A, B, and let [phi] be the infinitely small angle between AB and A´B´. Then if [alpha], [beta] be the orthogonal projections of A´, B´ on AB, we have

A[alpha] - B[beta] = AB - [alpha][beta] = AB(1 - cos [phi]) = ½AB·[phi]²,

ultimately. Since this is of the second order, the products F·A[alpha] and F·B[beta] are ultimately equal.

The total work done by two concurrent forces acting on a particle, or on a rigid body, in any infinitely small displacement, is equal to the work of their resultant. Let AB, AC (fig. 46) represent the forces, AD their resultant, and let AH be the direction of the displacement [delta]s of the point A. The proposition follows at once from the fact that the sum of orthogonal projections of [->AB], [->AC] on AH is equal to the projection of [->AD]. It is to be noticed that AH need not be in the same plane with AB, AC.

It follows from the preceding statements that any two systems of forces which are statically equivalent, according to the principles of §§ 4, 8, will (to the first order of small quantities) do the same amount of work in any infinitely small displacement of a rigid body to which they may be applied. It is also evident that the total work done in two or more successive infinitely small displacements is equal to the work done in the resultant displacement.

The work of a couple in any infinitely small rotation of a rigid body about an axis perpendicular to the plane of the couple is equal to the product of the moment of the couple into the angle of rotation, proper conventions as to sign being observed. Let the couple consist of two forces P, P (fig. 47) in the plane of the paper, and let J be the point where this plane is met by the axis of rotation. Draw JBA perpendicular to the lines of action, and let [epsilon] be the angle of rotation. The work of the couple is

P·JA·[epsilon] - P·JB·[epsilon] = P·AB·[epsilon] = G[epsilon],

if G be the moment of the couple.

The analytical calculation of the work done by a system of forces in any infinitesimal displacement is as follows. For a two-dimensional system we have, in the notation of §§ 3, 4,

[Sigma](X[delta]x + Y[delta]y) = [Sigma]{X([lambda] - y[epsilon]) + Y([mu] + x[epsilon])} = [Sigma](X)·[lambda] + [Sigma](Y)·[mu] + [Sigma](xY - yX)[epsilon] = X[lambda] + Y[mu] + N[epsilon]. (1)

Again, for a three-dimensional system, in the notation of §§ 7, 8,

[Sigma](X[delta]x + Y[delta]y + Z[delta]z) = [Sigma]{(X([lambda] + [eta]z - [zeta]y) + Y([mu] + [zeta]x - [xi]x) + Z([nu] + [xi]y - [eta]x)} = [Sigma](X)·[lambda] + [Sigma](Y)·[mu] + [Sigma](Z)·[nu] + [Sigma](yZ - zY)·[xi] + [Sigma](zX - xZ)·[eta] + [Sigma](xY - yX)·[zeta] = X[lambda] + Y[mu] + Z[nu] + L[xi] + M[eta] + N[zeta]. (2)

This expression gives the work done by a given wrench when the body receives a given infinitely small twist; it must of course be an absolute invariant for all transformations of rectangular axes. The first three terms express the work done by the components of a force (X, Y, Z) acting at O, and the remaining three terms express the work of a couple (L, M, N).

The work done by a wrench about a given screw, when the body twists about a second given screw, may be calculated directly as follows. In fig. 48 let R, G be the force and couple of the wrench, [epsilon],[tau] the rotation and translation in the twist. Let the axes of the wrench and the twist be inclined at an angle [theta], and let h be the shortest distance between them. The displacement of the point H in the figure, resolved in the direction of R, is [tau] cos [theta] - [epsilon]h sin [theta]. The work is therefore

R([tau] cos [theta] - [epsilon]h sin [theta]) + G cos [theta] = R[epsilon]{(p + p´) cos [theta] - h sin [theta]}, (3)

if G = pR, [tau] = p´[epsilon], i.e. p, p´ are the pitches of the two screws. The factor (p + p´) cos[theta] - h sin[theta] is called the _virtual coefficient_ of the two screws which define the types of the wrench and twist, respectively.

A screw is determined by its axis and its pitch, and therefore involves five Independent elements. These may be, for instance, the five ratios [xi]:[eta]:[zeta]:[lambda]:[mu]:[nu] of the six quantities which specify an infinitesimal twist about the screw. If the twist is a pure rotation, these quantities are subject to the relation

[lambda][xi] + [mu][eta] + [nu][zeta] = 0. (4)

In the analytical investigations of line geometry, these six quantities, supposed subject to the relation (4), are used to specify a line, and are called the six "co-ordinates" of the line; they are of course equivalent to only four independent quantities. If a line is a null-line with respect to the wrench (X, Y, Z, L, M, N), the work done in an infinitely small rotation about it is zero, and its co-ordinates are accordingly subject to the further relation

L[xi] + M[eta] + N[zeta] + X[lambda] + Y[mu] + Z[nu] = 0, (5)

where the coefficients are constant. This is the equation of a "linear complex" (cf. § 8).

Two screws are _reciprocal_ when a wrench about one does no work on a body which twists about the other. The condition for this is

[lambda][xi]´ + [mu][eta]´ + [nu][zeta]´ + [lambda]´[xi] + [mu]´[eta] + [nu]´[zeta] = 0, (6)

if the screws be defined by the ratios [xi] : [eta] : [zeta] : [lambda] : [mu] : [nu] and [xi]´ : [eta]´ : [zeta]´ : [lambda]´ : [mu]´ : [nu]´, respectively. The theory of the screw-systems which are reciprocal to one, two, three, four given screws respectively has been investigated by Sir R. S. Ball.

Considering a rigid body in any given position, we may contemplate the whole group of infinitesimal displacements which might be given to it. If the extraneous forces are in equilibrium the total work which they would perform in any such displacement would be zero, since they reduce to a zero force and a zero couple. This is (in part) the celebrated principle of _virtual velocities_, now often described as the principle of _virtual work_, enunciated by John Bernoulli (1667-1748). The word "virtual" is used because the displacements in question are not regarded as actually taking place, the body being in fact at rest. The "velocities" referred to are the velocities of the various points of the body in any imagined motion of the body through the position in question; they obviously bear to one another the same ratios as the corresponding infinitesimal displacements. Conversely, we can show that if the virtual work of the extraneous forces be zero for every infinitesimal displacement of the body as rigid, these forces must be in equilibrium. For by giving the body (in imagination) a displacement of translation we learn that the sum of the resolved parts of the forces in any assigned direction is zero, and by giving it a displacement of pure rotation we learn that the sum of the moments about any assigned axis is zero. The same thing follows of course from the analytical expression (2) for the virtual work. If this vanishes for all values of [lambda], [mu], [nu], [xi], [eta], [zeta] we must have X, Y, Z, L, M, N = 0, which are the conditions of equilibrium.

The principle can of course be extended to any system of particles or rigid bodies, connected together in any way, provided we take into account the internal stresses, or reactions, between the various parts. Each such reaction consists of two equal and opposite forces, both of which may contribute to the equation of virtual work.

The proper significance of the principle of virtual work, and of its converse, will appear more clearly when we come to kinetics (§ 16); for the present it may be regarded merely as a compact and (for many purposes) highly convenient summary of the laws of equilibrium. Its special value lies in this, that by a suitable adjustment of the hypothetical displacements we are often enabled to eliminate unknown reactions. For example, in the case of a particle lying on a smooth curve, or on a smooth surface, if it be displaced along the curve, or on the surface, the virtual work of the normal component of the pressure may be ignored, since it is of the second order. Again, if two bodies are connected by a string or rod, and if the hypothetical displacements be adjusted so that the distance between the points of attachment is unaltered, the corresponding stress may be ignored. This is evident from fig. 45; if AB, A´B´ represent the two positions of a string, and T be the tension, the virtual work of the two forces ±T at A, B is T(A[alpha] - B[beta]), which was shown to be of the second order. Again, the normal pressure between two surfaces disappears from the equation, provided the displacements be such that one of these surfaces merely slides relatively to the other. It is evident, in the first place, that in any displacement common to the two surfaces, the work of the two equal and opposite normal pressures will cancel; moreover if, one of the surfaces being fixed, an infinitely small displacement shifts the point of contact from A to B, and if A´ be the new position of that point of the sliding body which was at A, the projection of AA´ on the normal at A is of the second order. It is to be noticed, in this case, that the tangential reaction (if any) between the two surfaces is not eliminated. Again, if the displacements be such that one curved surface rolls without sliding on another, the reaction, whether normal or tangential, at the point of contact may be ignored. For the virtual work of two equal and opposite forces will cancel in any displacement which is common to the two surfaces; whilst, if one surface be fixed, the displacement of that point of the rolling surface which was in contact with the other is of the second order. We are thus able to imagine a great variety of mechanical systems to which the principle of virtual work can be applied without any regard to the internal stresses, provided the hypothetical displacements be such that none of the connexions of the system are violated.

If the system be subject to gravity, the corresponding part of the virtual work can be calculated from the displacement of the centre of gravity. If W1, W2, ... be the weights of a system of particles, whose depths below a fixed horizontal plane of reference are z1, z2, ..., respectively, the virtual work of gravity is

W1[delta]·z1 + W2[delta]z2 + ... = [delta](W1z1 + W2z2 + ...) (7) = (W1 + W2 + ...) [delta][|z],

where [|z] is the depth of the centre of gravity (see § 8 (14) and § 11 (6)). This expression is the same as if the whole mass were concentrated at the centre of gravity, and displaced with this point. An important conclusion is that in any displacement of a system of bodies in equilibrium, such that the virtual work of all forces except gravity may be ignored, the depth of the centre of gravity is "stationary."

The question as to stability of equilibrium belongs essentially to kinetics; but we may state by anticipation that in cases where gravity is the only force which does work, the equilibrium of a body or system of bodies is stable only if the depth of the centre of gravity be a maximum.

Consider, for instance, the case of a bar resting with its ends on two smooth inclines (fig. 18). If the bar be displaced in a vertical plane so that its ends slide on the two inclines, the instantaneous centre is at the point J. The displacement of G is at right angles to JG; this shows that for equilibrium JG must be vertical. Again, the locus of G is an arc of an ellipse whose centre is in the intersection of the planes; since this arc is convex upwards the equilibrium is unstable. A general criterion for the case of a rigid body movable in two dimensions, with one degree of freedom, can be obtained as follows. We have seen (§ 3) that the sequence of possible positions is obtained if we imagine the "body-centrode" to roll on the "space-centrode." For equilibrium, the altitude of the centre of gravity G must be stationary; hence G must lie in the same vertical line with the point of contact J of the two curves. Further, it is known from the theory of "roulettes" that the locus of G will be concave or convex upwards according as

cos[phi] 1 1 -------- = ----- + ------, (8) h [rho] [rho]´

where [rho], [rho]´ are the radii of curvature of the two curves at J, [phi] is the inclination of the common tangent at J to the horizontal, and h is the height of G above J. The signs of [rho], [rho]´ are to be taken positive when the curvatures are as in the standard case shown in fig. 49. Hence for stability the upper sign must obtain in (8). The same criterion may be arrived at in a more intuitive manner as follows. If the body be supposed to roll (say to the right) until the curves touch at J´, and if JJ´ = [delta]s, the angle through which the upper figure rotates is [delta]s/[rho] + [delta]s/[rho]´, and the horizontal displacement of G is equal to the product of this expression into h. If this displacement be less than the horizontal projection of JJ´, viz. [delta]s cos[phi], the vertical through the new position of G will fall to the left of J´ and gravity will tend to restore the body to its former position. It is here assumed that the remaining forces acting on the body in its displaced position have zero moment about J´; this is evidently the case, for instance, in the problem of "rocking stones."

The principle of virtual work is specially convenient in the theory of frames (§ 6), since the reactions at smooth joints and the stresses in inextensible bars may be left out of account. In particular, in the case of a frame which is just rigid, the principle enables us to find the stress in any one bar independently of the rest. If we imagine the bar in question to be removed, equilibrium will still persist if we introduce two equal and opposite forces S, of suitable magnitude, at the joints which it connected. In any infinitely small deformation of the frame as thus modified, the virtual work of the forces S, together with that of the original extraneous forces, must vanish; this determines S.

As a simple example, take the case of a light frame, whose bars form the slides of a rhombus ABCD with the diagonal BD, suspended from A and carrying a weight W at C; and let it be required to find the stress in BD. If we remove the bar BD, and apply two equal and opposite forces S at B and D, the equation is

W·[delta](2l cos[theta]) + 2S·[delta](l sin [theta]) = 0,

where l is the length of a side of the rhombus, and [theta] its inclination to the vertical. Hence

S = W tan [theta] = W·BD/AC. (8)

The method is specially appropriate when the frame, although just rigid, is not "simple" in the sense of § 6, and when accordingly the method of reciprocal figures is not immediately available. To avoid the intricate trigonometrical calculations which would often be necessary, graphical devices have been introduced by H. Müller-Breslau and others. For this purpose the infinitesimal displacements of the various joints are replaced by finite lengths proportional to them, and therefore proportional to the velocities of the joints in some imagined motion of the deformable frame through its actual configuration; this is really (it may be remarked) a reversion to the original notion of "virtual velocities." Let J be the instantaneous centre for any bar CD (fig. 12), and let s1, s2 represent the virtual velocities of C, D. If these lines be turned through a right angle in the same sense, they take up positions such as CC´, DD´, where C´, D´ are on JC, JD, respectively, and C´D´ is parallel to CD. Further, if F1 (fig. 51) be any force acting on the joint C, its virtual work will be equal to the moment of F1 about C´; the equation of virtual work is thus transformed into an equation of moments.

Consider, for example, a frame whose sides form the six sides of a hexagon ABCDEF and the three diagonals AD, BE, CF; and suppose that it is required to find the stress in CF due to a given system of extraneous forces in equilibrium, acting on the joints. Imagine the bar CF to be removed, and consider a deformation in which AB is fixed. The instantaneous centre of CD will be at the intersection of AD, BC, and if C´D´ be drawn parallel to CD, the lines CC´, DD´ may be taken to represent the virtual velocities of C, D turned each through a right angle. Moreover, if we draw D´E´ parallel to DE, and E´F´ parallel to EF, the lines CC´, DD´, EE´, FF´ will represent on the same scale the virtual velocities of the points C, D, E, F, respectively, turned each through a right angle. The equation of virtual work is then formed by taking moments about C´, D´, E´, F´ of the extraneous forces which act at C, D, E, F, respectively. Amongst these forces we must include the two equal and opposite forces S which take the place of the stress in the removed bar FC.

The above method lends itself naturally to the investigation of the _critical forms_ of a frame whose general structure is given. We have seen that the stresses produced by an equilibrating system of extraneous forces in a frame which is just rigid, according to the criterion of § 6, are in general uniquely determinate; in particular, when there are no extraneous forces the bars are in general free from stress. It may however happen that owing to some special relation between the lengths of the bars the frame admits of an infinitesimal deformation. The simplest case is that of a frame of three bars, when the three joints A, B, C fall into a straight line; a small displacement of the joint B at right angles to AC would involve changes in the lengths of AB, BC which are only of the second order of small quantities. Another example is shown in fig. 53. The graphical method leads at once to the detection of such cases. Thus in the hexagonal frame of fig. 52, if an infinitesimal deformation is possible without removing the bar CF, the instantaneous centre of CF (when AB is fixed) will be at the intersection of AF and BC, and since CC´, FF´ represent the virtual velocities of the points C, F, turned each through a right angle, C´F´ must be parallel to CF. Conversely, if this condition be satisfied, an infinitesimal deformation is possible. The result may be generalized into the statement that a frame has a critical form whenever a frame of the same structure can be designed with corresponding bars parallel, but without complete geometric similarity. In the case of fig. 52 it may be shown that an equivalent condition is that the six points A, B, C, D, E, F should lie on a conic (M. W. Crofton). This is fulfilled when the opposite sides of the hexagon are parallel, and (as a still more special case) when the hexagon is regular.

When a frame has a critical form it may be in a state of stress independently of the action of extraneous forces; moreover, the stresses due to extraneous forces are indeterminate, and may be infinite. For suppose as before that one of the bars is removed. If there are no extraneous forces the equation of virtual work reduces to S·[delta]s = 0, where S is the stress in the removed bar, and [delta]s is the change in the distance between the joints which it connected. In a critical form we have [delta]s = 0, and the equation is satisfied by an arbitrary value of S; a consistent system of stresses in the remaining bars can then be found by preceding rules. Again, when extraneous forces P act on the joints, the equation is

[Sigma](P·[delta]p) + S·[delta]s = 0,

where [delta]p is the displacement of any joint in the direction of the corresponding force P. If [Sigma](P·[delta]p) = 0, the stresses are merely indeterminate as before; but if [Sigma] (P·[delta]p) does not vanish, the equation cannot be satisfied by any finite value of S, since [delta]s = 0. This means that, if the material of the frame were absolutely unyielding, no finite stresses in the bars would enable it to withstand the extraneous forces. With actual materials, the frame would yield elastically, until its configuration is no longer "critical." The stresses in the bars would then be comparatively very great, although finite. The use of frames which approximate to a critical form is of course to be avoided in practice.

A brief reference must suffice to the theory of three dimensional frames. This is important from a technical point of view, since all structures are practically three-dimensional. We may note that a frame of n joints which is just rigid must have 3n - 6 bars; and that the stresses produced in such a frame by a given system of extraneous forces in equilibrium are statically determinate, subject to the exception of "critical forms."

§ 10. _Statics of Inextensible Chains._--The theory of bodies or structures which are deformable in their smallest parts belongs properly to elasticity (q.v.). The case of inextensible strings or chains is, however, so simple that it is generally included in expositions of pure statics.

It is assumed that the form can be sufficiently represented by a plane curve, that the stress (tension) at any point P of the curve, between the two portions which meet there, is in the direction of the tangent at P, and that the forces on any linear element [delta]s must satisfy the conditions of equilibrium laid down in § 1. It follows that the forces on any finite portion will satisfy the conditions of equilibrium which apply to the case of a rigid body (§ 4).

We will suppose in the first instance that the curve is plane. It is often convenient to resolve the forces on an element PQ (= [delta]s) in the directions of the tangent and normal respectively. If T, T + [delta]T be the tensions at P, Q, and [delta][psi] be the angle between the directions of the curve at these points, the components of the tensions along the tangent at P give (T + [delta]T) cos [psi] - T, or [delta]T, ultimately; whilst for the component along the normal at P we have (T + [delta]T) sin [delta][psi], or T[delta][psi], or T[delta]s/[rho], where [rho] is the radius of curvature.

Suppose, for example, that we have a light string stretched over a smooth curve; and let R[delta]s denote the normal pressure (outwards from the centre of curvature) on [delta]s. The two resolutions give [delta]T = 0, T[delta][psi] = R[delta]s, or

T = const., R = T/[rho]. (1)

The tension is constant, and the pressure per unit length varies as the curvature.

Next suppose that the curve is "rough"; and let F[delta]s be the tangential force of friction on [delta]s. We have [delta]T ± F[delta]s = 0, T[delta][psi] = R[delta]s, where the upper or lower sign is to be taken according to the sense in which F acts. We assume that in limiting equilibrium we have F = [mu]R, everywhere, where [mu] is the coefficient of friction. If the string be on the point of slipping in the direction in which [psi] increases, the lower sign is to be taken; hence [delta]T = F[delta]s = [mu]T[delta][psi], whence

T = T0 e^([mu][psi]), (2)

if T0 be the tension corresponding to [psi] = 0. This illustrates the resistance to dragging of a rope coiled round a post; e.g. if we put [mu] = .3, [psi] = 2[pi], we find for the change of tension in one turn T/T0 = 6.5. In two turns this ratio is squared, and so on.

Again, take the case of a string under gravity, in contact with a smooth curve in a vertical plane. Let [psi] denote the inclination to the horizontal, and w [delta]s the weight of an element [delta]s. The tangential and normal components of w[delta]s are -s sin [psi] and -w [delta]s cos [psi]. Hence

[delta]T = w [delta]s sin [psi], T [delta][psi] = w [delta]s cos [psi] + R[delta]s. (3)

If we take rectangular axes Ox, Oy, of which Oy is drawn vertically upwards, we have [delta]y = sin[psi] [delta]s, whence [delta]T = w[delta]y. If the string be uniform, w is constant, and

T = wy + const. = w(y - y0), (4)

say; hence the tension varies as the height above some fixed level (y0). The pressure is then given by the formula

d[psi] R = T ------ - w cos [psi]. (5) ds

In the case of a chain hanging freely under gravity it is usually convenient to formulate the conditions of equilibrium of a finite portion PQ. The forces on this reduce to three, viz. the weight of PQ and the tensions at P, Q. Hence these three forces will be concurrent, and their ratios will be given by a triangle of forces. In particular, if we consider a length AP beginning at the lowest point A, then resolving horizontally and vertically we have

T cos [psi] = T0, T sin [psi] = W, (6)

where T0 is the tension at A, and W is the weight of PA. The former equation expresses that the horizontal tension is constant.

If the chain be uniform we have W = ws, where s is the arc AP: hence ws = T0 tan[psi]. If we write T0 = wa, so that a is the length of a portion of the chain whose weight would equal the horizontal tension, this becomes

s = a tan [psi]. (7)

This is the "intrinsic" equation of the curve. If the axes of x and y be taken horizontal and vertical (upwards), we derive

x = a log (sec [psi] + tan [psi]), y = a sec [psi]. (8)

Eliminating [psi] we obtain the Cartesian equation

x y = a cosh --- (9) a

of the _common catenary_, as it is called (fig. 56). The omission of the additive arbitrary constants of integration in (8) is equivalent to a special choice of the origin O of co-ordinates; viz. O is at a distance a vertically below the lowest point ([psi] = 0) of the curve. The horizontal line through O is called the _directrix_. The relations

s = a sinh x/a, y² = a² + s², T = T0 sec [psi] = wy, (10)

which are involved in the preceding formulae are also noteworthy. It is a classical problem in the calculus of variations to deduce the equation (9) from the condition that the depth of the centre of gravity of a chain of given length hanging between fixed points must be stationary (§ 9). The length a is called the _parameter_ of the catenary; it determines the scale of the curve, all catenaries being geometrically similar. If weights be suspended from various points of a hanging chain, the intervening portions will form arcs of equal catenaries, since the horizontal tension (wa) is the same for all. Again, if a chain pass over a perfectly smooth peg, the catenaries in which it hangs on the two sides, though usually of different parameters, will have the same directrix, since by (10) y is the same for both at the peg.

As an example of the use of the formulae we may determine the maximum span for a wire of given material. The condition is that the tension must not exceed the weight of a certain length [lambda] of the wire. At the ends we shall have y = [lambda], or

x [lambda] = a cosh ---, (11) a

and the problem is to make x a maximum for variations of a. Differentiating (11) we find that, if dx/da = 0,

x x --- tanh --- = 1. (12) a a

It is easily seen graphically, or from a table of hyperbolic tangents, that the equation u tanh u = 1 has only one positive root (u = 1.200); the span is therefore

2x = 2au = 2[lambda]/sinh u = 1.326[lambda],

and the length of wire is

2s = 2[lambda]/u = 1.667 [lambda].

The tangents at the ends meet on the directrix, and their inclination to the horizontal is 56° 30´.

The relation between the sag, the tension, and the span of a wire (e.g. a telegraph wire) stretched nearly straight between two points A, B at the same level is determined most simply from first principles. If T be the tension, W the total weight, k the sag in the middle, and [psi] the inclination to the horizontal at A or B, we have 2T[psi] = W, AB = 2[rho][psi], approximately, where [rho] is the radius of curvature. Since 2k[rho] = (½AB)², ultimately, we have

k = (1/8)W·AB/T. (13)

The same formula applies if A, B be at different levels, provided k be the sag, measured vertically, half way between A and B.

In relation to the theory of suspension bridges the case where the weight of any portion of the chain varies as its horizontal projection is of interest. The vertical through the centre of gravity of the arc AP (see fig. 55) will then bisect its horizontal projection AN; hence if PS be the tangent at P we shall have AS = SN. This property is characteristic of a parabola whose axis is vertical. If we take A as origin and AN as axis of x, the weight of AP may be denoted by wx, where w is the weight per unit length at A. Since PNS is a triangle of forces for the portion AP of the chain, we have wx/T0 = PN/NS, or

y = w·x²/2T0, (14)

which is the equation of the parabola in question. The result might of course have been inferred from the theory of the parabolic funicular in § 2.

Finally, we may refer to the _catenary of uniform strength_, where the cross-section of the wire (or cable) is supposed to vary as the tension. Hence w, the weight per foot, varies as T, and we may write T = w[lambda], where [lambda] is a constant length. Resolving along the normal the forces on an element [delta]s, we find T[delta][psi] = w[delta]s cos[psi], whence

ds p = ------ = [lambda] sec [psi]. (15) d[psi]

From this we derive

x x = [lambda][psi], y = [lambda] log sec --------, (16) [lambda]

where the directions of x and y are horizontal and vertical, and the origin is taken at the lowest point. The curve (fig. 58) has two vertical asymptotes x = ± ½[pi][lambda]; this shows that however the thickness of a cable be adjusted there is a limit [pi][lambda] to the horizontal span, where [lambda] depends on the tensile strength of the material. For a uniform catenary the limit was found above to be 1.326[lambda].

For investigations relating to the equilibrium of a string in three dimensions we must refer to the textbooks. In the case of a string stretched over a smooth surface, but in other respects free from extraneous force, the tensions at the ends of a small element [delta]s must be balanced by the normal reaction of the surface. It follows that the osculating plane of the curve formed by the string must contain the normal to the surface, i.e. the curve must be a "geodesic," and that the normal pressure per unit length must vary as the principal curvature of the curve.

§ 11. _Theory of Mass-Systems._--This is a purely geometrical subject. We consider a system of points P1, P2 ..., P_n, with which are associated certain coefficients m1, m2, ... m_n, respectively. In the application to mechanics these coefficients are the masses of particles situate at the respective points, and are therefore all positive. We shall make this supposition in what follows, but it should be remarked that hardly any difference is made in the theory if some of the coefficients have a different sign from the rest, except in the special case where [Sigma](m) = 0. This has a certain interest in magnetism.

In a given mass-system there exists one and only one point G such that

[Sigma](m·[->GP]) = 0. (1)

For, take any point O, and construct the vector

[Sigma](m·[->OP]) [->OG] = -----------------. (2) [Sigma](m)

Then

[Sigma](m·[->GP]) = [Sigma]{m([->GO] + [->OP])} = [Sigma](m)·[->GO] + [Sigma](m)·[->OP] = 0. (3)

Also there cannot be a distinct point G´ such that [Sigma](m·G´P) = 0, for we should have, by subtraction,

[Sigma]{m([->GP] + [->PG´])} = 0, or [Sigma](m)·GG´ = 0; (4)

i.e. G´ must coincide with G. The point G determined by (1) is called the _mass-centre_ or _centre of inertia_ of the given system. It is easily seen that, in the process of determining the mass-centre, any group of particles may be replaced by a single particle whose mass is equal to that of the group, situate at the mass-centre of the group.

If through P1, P2, ... P_n we draw any system of parallel planes meeting a straight line OX in the points M1, M2 ... M_n, the collinear vectors [->OM1], [->OM2] ... [->OM_n] may be called the "projections" of [->OP1], [->OP2], ... [->OP_n] on OX. Let these projections be denoted algebraically by x1, x2, ... x_n, the sign being positive or negative according as the direction is that of OX or the reverse. Since the projection of a vector-sum is the sum of the projections of the several vectors, the equation (2) gives

[Sigma](mx) [|x] = -----------, (5) [Sigma](m)

if [|x] be the projection of [->OG]. Hence if the Cartesian co-ordinates of P1, P2, ... P_n relative to any axes, rectangular or oblique be (x1, y1, z1), (x2, y2, z2), ..., (x_n, y_n, z_n), the mass-centre ([|x], [|y], [|z]) is determined by the formulae

[Sigma](mx) [Sigma](my) [Sigma](mz) [|x] = -----------, [|y] = -----------, [|z] = -----------. (6) [Sigma](m) [Sigma](m) [Sigma](m)

If we write x = [|x] + [xi], y = [|y] + [eta], z = [|z] + [zeta], so that [xi], [eta], [zeta] denote co-ordinates relative to the mass-centre G, we have from (6)

[Sigma](m[xi]) = 0, [Sigma](m[eta]) = 0, [Sigma](m[zeta]) = 0. (7)

One or two special cases may be noticed. If three masses [alpha], [beta], [gamma] be situate at the vertices of a triangle ABC, the mass-centre of [beta] and [gamma] is at a point A´ in BC, such that [beta]·BA´ = [gamma]·A´C. The mass-centre (G) of [alpha], [beta], [gamma] will then divide AA´ so that [alpha]·AG = ([beta] + [gamma]) GA´. It is easily proved that

[alpha] : [beta] : [gamma] = [Delta]BGA : [Delta]GCA : [Delta]GAB;

also, by giving suitable values (positive or negative) to the ratios [alpha] : [beta] : [gamma] we can make G assume any assigned position in the plane ABC. We have here the origin of the "barycentric co-ordinates" of Möbius, now usually known as "areal" co-ordinates. If [alpha] + [beta] + [gamma] = 0, G is at infinity; if [alpha] = [beta] = [gamma], G is at the intersection of the median lines of the triangle; if [alpha] : [beta] : [gamma] = a : b : c, G is at the centre of the inscribed circle. Again, if G be the mass-centre of four particles [alpha], [beta], [gamma], [delta] situate at the vertices of a tetrahedron ABCD, we find

[alpha] : [beta] : [gamma] : [delta] = tet^n GBCD : tet^n GCDA : tet^n GDAB : tet^n GABC,

and by suitable determination of the ratios on the left hand we can make G assume any assigned position in space. If [alpha] + [beta] + [gamma] + [delta] = O, G is at infinity; if [alpha] = [beta] = [gamma] = [delta], G bisects the lines joining the middle points of opposite edges of the tetrahedron ABCD; if [alpha] : [beta] : [gamma] : [delta] = [Delta]BCD : [Delta]CDA : [Delta]DAB : [Delta]ABC, G is at the centre of the inscribed sphere.

If we have a continuous distribution of matter, instead of a system of discrete particles, the summations in (6) are to be replaced by integrations. Examples will be found in textbooks of the calculus and of analytical statics. As particular cases: the mass-centre of a uniform thin triangular plate coincides with that of three equal particles at the corners; and that of a uniform solid tetrahedron coincides with that of four equal particles at the vertices. Again, the mass-centre of a uniform solid right circular cone divides the axis in the ratio 3 : 1; that of a uniform solid hemisphere divides the axial radius in the ratio 3 : 5.

It is easily seen from (6) that if the configuration of a system of particles be altered by "homogeneous strain" (see ELASTICITY) the new position of the mass-centre will be at that point of the strained figure which corresponds to the original mass-centre.

The formula (2) shows that a system of concurrent forces represented by m1·[->OP1], m2·[->OP2], ... m_n·[->OP_n] will have a resultant represented hy [Sigma](m)·[->OG]. If we imagine O to recede to infinity in any direction we learn that a system of parallel forces proportional to m1, m2,... m_n, acting at P1, P2 ... P_n have a resultant proportional to [Sigma](m) which acts always through a point G fixed relatively to the given mass-system. This contains the theory of the "centre of gravity" (§§ 4, 9). We may note also that if P1, P2, ... P_n, and P1´, P2´, ... P_n´ represent two configurations of the series of particles, then

[Sigma](m·[->PP´]) = Sigma(m)·[->GG´], (8)

where G, G´ are the two positions of the mass-centre. The forces m1·[->P1P1´], m2·[->P2P2´], ... m_n·[->P_nP_n´], considered as localized vectors, do not, however, as a rule reduce to a single resultant.

We proceed to the theory of the _plane_, _axial_ and _polar quadratic moments_ of the system. The axial moments have alone a dynamical significance, but the others are useful as subsidiary conceptions. If h1, h2, ... h_n be the perpendicular distances of the particles from any fixed plane, the sum [Sigma](mh²) is the quadratic moment with respect to the plane. If p1, p2, ... p_n be the perpendicular distances from any given axis, the sum [Sigma](mp²) is the quadratic moment with respect to the axis; it is also called the _moment of inertia_ about the axis. If r1, r2, ... r_n be the distances from a fixed point, the sum [Sigma](mr²) is the quadratic moment with respect to that point (or pole). If we divide any of the above quadratic moments by the total mass [Sigma](m), the result is called the _mean square_ of the distances of the particles from the respective plane, axis or pole. In the case of an axial moment, the square root of the resulting mean square is called the _radius of gyration_ of the system about the axis in question. If we take rectangular axes through any point O, the quadratic moments with respect to the co-ordinate planes are

I_x = [Sigma](mx²), I_y = [Sigma](my²), I_z = [Sigma](mz²); (9)

those with respect to the co-ordinate axes are

I_yz = [Sigma]{m(y² + z²)}, I_zx = [Sigma]{m(z² + x²)}, I_xy = [Sigma]{m(x² + y²)}; (10)

whilst the polar quadratic moment with respect to O is

I0 = [Sigma]{m(x² + y² + z²)}. (11)

We note that

I_yz = I_y + I_z, I_zx = I_z + I_x, I_xy = I_x + I_y, (12)

and

I0 = I_x + I_y + I_z = ½(I_yz + I_zx + I_xy). (13)

In the case of continuous distributions of matter the summations in (9), (10), (11) are of course to be replaced by integrations. For a uniform thin circular plate, we find, taking the origin at its centre, and the axis of z normal to its plane, I0 = ½Ma², where M is the mass and a the radius. Since I_x = I_y, I_z = 0, we deduce I_zx = ½Ma², I_xy = ½Ma²; hence the value of the squared radius of gyration is for a diameter ¼a², and for the axis of symmetry ½a². Again, for a uniform solid sphere having its centre at the origin we find I0 = (3/5)Ma², I_x = I_y = I_z = (1/5)Ma², I_yz = I_zx = l_xy = (3/5)Ma²; i.e. the square of the radius of gyration with respect to a diameter is (2/5)a². The method of homogeneous strain can be applied to deduce the corresponding results for an ellipsoid of semi-axes a, b, c. If the co-ordinate axes coincide with the principal axes, we find I_x = (1/5)Ma², I_y = (1/5)Mb², I_z = (1/5)Mc², whence I_yz = (1/5)M (b² + c²), &c.

If [phi](x, y, z) be any homogeneous quadratic function of x, y, z, we have

[Sigma]{m[phi](x, y, z)} = [Sigma] {m[phi]([|x] + [xi], [|y] + [eta], [|z] + [zeta])} = [Sigma] {m[phi](x, y, z)} + [Sigma]{m[phi]([xi], [eta], [zeta])}, (14)

since the terms which are bilinear in respect to [|x], [|y], [|z], and [xi], [eta], [zeta] vanish, in virtue of the relations (7). Thus

I_x = I[xi] + [Sigma](m)x², (15)

I_yz = I[eta][zeta] + [Sigma](m)·(y² + z²), (16)

with similar relations, and

I_O = I_G + [Sigma](m)·OG². (17)

The formula (16) expresses that the squared radius of gyration about any axis (Ox) exceeds the squared radius of gyration about a parallel axis through G by the square of the distance between the two axes. The formula (17) is due to J. L. Lagrange; it may be written

[Sigma](m·OP²) [Sigma](m·GP²) -------------- = -------------- + OG², (18) [Sigma](m) [Sigma](m)

and expresses that the mean square of the distances of the particles from O exceeds the mean square of the distances from G by OG². The mass-centre is accordingly that point the mean square of whose distances from the several particles is least. If in (18) we make O coincide with P1, P2, ... P_n in succession, we obtain

0 + m2·P1P2² + ... + mn·P1P_n² = [Sigma](m·GP²) + [Sigma](m)·GP1², \ m1·P2P1² + 0 + ... + mn·P2P_n² = [Sigma](m·GP²) + [Sigma](m)·GP2², > (19) ... ... ... ... ... | m1·P_nP1² + m2·P_nP2² + ... + 0 = [Sigma](m·GP²) + [Sigma](m)·GP_n². /

If we multiply these equations by m1, m2 ... m_n, respectively, and add, we find

[Sigma][Sigma](m_r m_s·P_r P_s²) = [Sigma](m)·[Sigma](m·GP²), (20)

provided the summation [Sigma][Sigma] on the left hand be understood to include each pair of particles once only. This theorem, also due to Lagrange, enables us to express the mean square of the distances of the particles from the centre of mass in terms of the masses and mutual distances. For instance, considering four equal particles at the vertices of a regular tetrahedron, we can infer that the radius R of the circumscribing sphere is given by R² = (3/8)a², if a be the length of an edge.

Another type of quadratic moment is supplied by the _deviation-moments_, or _products of inertia_ of a distribution of matter. Thus the sum [Sigma](m·yz) is called the "product of inertia" with respect to the planes y = 0, z = 0. This may be expressed In terms of the product of inertia with respect to parallel planes through G by means of the formula (14); viz.:--

[Sigma](m·yz) = [Sigma](m·[eta][zeta]) + [Sigma](m)·yz (21)

The quadratic moments with respect to different planes through a fixed point O are related to one another as follows. The moment with respect to the plane

[lambda]x + [mu]y + [nu]z = 0, (22)

where [lambda], [mu], [nu] are direction-cosines, is

[Sigma]{(m([lambda]x + [mu]y + [nu]z)²} = [Sigma](mx²)·[lambda]² + [Sigma](my²)·[mu]² + [Sigma](mz²)·[nu]² + 2[Sigma](myz)·[mu][nu] + 2[Sigma](mzx)·[nu][lambda] + 2[Sigma](mxy)·[lambda][mu], (23)

and therefore varies as the square of the perpendicular drawn from O to a tangent plane of a certain quadric surface, the tangent plane in question being parallel to (22). If the co-ordinate axes coincide with the principal axes of this quadric, we shall have

[Sigma](myz) = 0, [Sigma](mzx) = 0, [Sigma](mxy) = 0; (24)

and if we write

[Sigma](mx²) = Ma², [Sigma](my²) = Mb², [Sigma](mz²) = Mc², (25)

where M = [Sigma](m), the quadratic moment becomes M(a²[lambda]² + b²[mu]² + c²[nu]²), or Mp², where p is the distance of the origin from that tangent plane of the ellipsoid

x² y² z² --- + --- + --- = 1, (26) a² b² c²

which is parallel to (22). It appears from (24) that through any assigned point O three rectangular axes can be drawn such that the product of inertia with respect to each pair of co-ordinate planes vanishes; these are called the _principal axes of inertia_ at O. The ellipsoid (26) was first employed by J. Binet (1811), and may be called "Binet's Ellipsoid" for the point O. Evidently the quadratic moment for a variable plane through O will have a "stationary" value when, and only when, the plane coincides with a principal plane of (26). It may further be shown that if Binet's ellipsoid be referred to any system of conjugate diameters as co-ordinate axes, its equation will be

x´² y´² z´² --- + --- + --- = 1, (27) a´² b´² c´²

provided

[Sigma](mx´²) = Ma´², [Sigma](my´²) Mb´², [Sigma](mz´²) = Mc´²;

also that

[Sigma](my´z´) = 0, [Sigma](mz´x´) = 0, [Sigma](mx´y´) = 0. (28)

Let us now take as co-ordinate axes the principal axes of inertia at the mass-centre G. If a, b, c be the semi-axes of the Binet's ellipsoid of G, the quadratic moment with respect to the plane [lambda]x + [mu]y + [nu]z = 0 will be M(a²[lambda]² + b²[mu]² + c²[nu]²), and that with respect to a parallel plane

[lambda]x + [mu]y + [nu]z = p (29)

will be M(a²[lambda]² + b²[mu]² + c²[nu]² + p²), by (15). This will have a given value Mk², provided

p² = (k² - a²)[lambda]² + (k² - b²)[mu]² + (k² - c²)[nu]². (30)

Hence the planes of constant quadratic moment Mk² will envelop the quadric

x² y² z² ------- + ------- + ------- = 1, (31) k² - a² k² - b² k² - c²

and the quadrics corresponding to different values of k² will be confocal. If we write

k² = a² + b² + c² + [theta], b² + c² = [alpha]², c² + a² = [beta]², a² + b² = [gamma]² (32)

the equation (31) becomes

x² y² z² ------------------ + ----------------- + ------------------ = 1 (33) [alpha]² + [theta] [beta]² + [theta] [gamma]² + [theta]

for different values of [theta] this represents a system of quadrics confocal with the ellipsoid

x² y² z² -------- + ------- + -------- = 1, (34) [alpha]² [beta]² [gamma]²

which we shall meet with presently as the "ellipsoid of gyration" at G. Now consider the tangent plane [omega] at any point P of a confocal, the tangent plane [omega]´ at an adjacent point N´, and a plane [omega]´´ through P parallel to [omega]´. The distance between the planes [omega]´ and [omega]´´ will be of the second order of small quantities, and the quadratic moments with respect to [omega]´ and [omega]´´ will therefore be equal, to the first order. Since the quadratic moments with respect to [omega] and [omega]´ are equal, it follows that [omega] is a plane of stationary quadratic moment at P, and therefore a principal plane of inertia at P. In other words, the principal axes of inertia at P arc the normals to the three confocals of the system (33) which pass through P. Moreover if x, y, z be the co-ordinates of P, (33) is an equation to find the corresponding values of [theta]; and if [theta]1, [theta]2, [theta]3 be the roots we find

[theta]1 + [theta]2 + [theta]3 = r² - [alpha]² - [beta]² -[gamma]², (35)

where r² = x² + y² + z². The squares of the radii of gyration about the principal axes at P may be denoted by k2² + k3², k3² + k1², k1² + k2²; hence by (32) and (35) they are r² - [theta]1, r² - [theta]2, r² - [theta]3, respectively.

To find the relations between the moments of inertia about different axes through any assigned point O, we take O as origin. Since the square of the distance of a point (x, y, z) from the axis

x y z -------- = ---- = ---- (36) [lambda] [mu] [nu]

is x² + y² + z² - ([lambda]x + [mu]y + [nu]z)², the moment of inertia about this axis is

I = [Sigma][m{([lambda]² + [mu]² + [nu]²)(x² + y² + z²) - ([lambda]x + [mu]y + [nu]z)²}] = A[lambda]² + B[mu]² + C[nu]² - 2F[mu][nu] - 2G[nu][lambda] - 2H[lambda][mu], (37)

provided

A = [Sigma]{m(y² + z²)}, B = [Sigma]{m(z² + x²)}, C = [Sigma]{m(x² + y²)}, F = [Sigma](myz), G = [Sigma](mzx), H = [Sigma](mxy); (38)

i.e. A, B, C are the moments of inertia about the co-ordinate axes, and F, G, H are the products of inertia with respect to the pairs of co-ordinate planes. If we construct the quadric

Ax² + By² + Cz² - 2Fyz - 2Gzx - 2Hxy = M[epsilon]^4 (39)

where [epsilon] is an arbitrary linear magnitude, the intercept r which it makes on a radius drawn in the direction [lambda], [mu], [nu] is found by putting x, y, z = [lambda]r, [mu]r, [nu]r. Hence, by comparison with (37),

I = M[epsilon]^4/r². (40)

The moment of inertia about any radius of the quadric (39) therefore varies inversely as the square of the length of this radius. When referred to its principal axes, the equation of the quadric takes the form

Ax² + By² + Cz² = M[epsilon]^4. (41)

The directions of these axes are determined by the property (24), and therefore coincide with those of the principal axes of inertia at O, as already defined in connexion with the theory of plane quadratic moments. The new A, B, C are called the _principal moments of inertia_ at O. Since they are essentially positive the quadric is an ellipsoid; it is called the _momental ellipsoid_ at O. Since, by (12), B + C > A, &c., the sum of the two lesser principal moments must exceed the greatest principal moment. A limitation is thus imposed on the possible forms of the momental ellipsoid; e.g. in the case of symmetry about an axis it appears that the ratio of the polar to the equatorial diameter of the ellipsoid cannot be less than 1/[root]2.

If we write A = M[alpha]², B = M[beta]², C = M[gamma]², the formula (37), when referred to the principal axes at O, becomes

I = M([alpha]²[lambda]² + [beta]²[mu]² + [gamma]²[nu]²) = Mp², (42)

if p denotes the perpendicular drawn from O in the direction ([lambda], [mu], [nu]) to a tangent plane of the ellipsoid

x² y² z² -------- + ------- + -------- = 1 (43) [alpha]² [beta]² [gamma]²

This is called the _ellipsoid of gyration_ at O; it was introduced into the theory by J. MacCullagh. The ellipsoids (41) and (43) are reciprocal polars with respect to a sphere having O as centre.

If A = B = C, the momental ellipsoid becomes a sphere; all axes through O are then principal axes, and the moment of inertia is the same for each. The mass-system is then said to possess kinetic symmetry about O.

If all the masses lie in a plane (z = 0) we have, in the notation of (25), c² = 0, and therefore A = Mb², B = Ma², C = M(a² + b²), so that the equation of the momental ellipsoid takes the form

b²x² + a²y² + (a² + b²)z² = [epsilon]^4. (44)

The section of this by the plane z = 0 is similar to

x² y² ---- + ---- = 1, (45) a² b²

which may be called the _momental ellipse_ at O. It possesses the property that the radius of gyration about any diameter is half the distance between the two tangents which are parallel to that diameter. In the case of a uniform triangular plate it may be shown that the momental ellipse at G is concentric, similar and similarly situated

to the ellipse which touches the sides of the triangle at their middle points.

The graphical methods of determining the moment of inertia of a plane system of particles with respect to any line in its plane may be briefly noticed. It appears from § 5 (fig. 31) that the linear moment of each particle about the line may be found by means of a funicular polygon. If we replace the mass of each particle by its moment, as thus found, we can in like manner obtain the quadratic moment of the system with respect to the line. For if the line in question be the axis of y, the first process gives us the values of mx, and the second the value of [Sigma](mx·x) or [Sigma](mx²). The construction of a second funicular may be dispensed with by the employment of a planimeter, as follows. In fig. 59 p is the line with respect to which moments are to be taken, and the masses of the respective particles are indicated by the corresponding segments of a line in the force-diagram, drawn parallel to p. The funicular ZABCD ... corresponding to any pole O is constructed for a system of forces acting parallel to p through the positions of the particles and proportional to the respective masses; and its successive sides are produced to meet p in the points H, K, L, M, ... As explained in § 5, the moment of the first particle is represented on a certain scale by HK, that of the second by KL, and so on. The quadratic moment of the first particle will then be represented by twice the area AHK, that of the second by twice the area BKL, and so on. The quadratic moment of the whole system is therefore represented by twice the area AHEDCBA. Since a quadratic moment is essentially positive, the various areas are to taken positive in all cases. If k be the radius of gyration about p we find

k² = 2 × area AHEDCBA × ON ÷ [alpha][beta],

where [alpha][beta] is the line in the force-diagram which represents the sum of the masses, and ON is the distance of the pole O from this line. If some of the particles lie on one side of p and some on the other, the quadratic moment of each set may be found, and the results added. This is illustrated in fig. 60, where the total quadratic moment is represented by the sum of the shaded areas. It is seen that for a given direction of p this moment is least when p passes through the intersection X of the first and last sides of the funicular; i.e. when p goes through the mass-centre of the given system; cf. equation (15).