iv. 226) has shown that success may be attained by a variety of

processes, including bichromated gelatin and the old bitumen process, and has investigated the effect of imperfect approximation during the exposure between the prepared plate and the original. For many purposes the copies, containing lines up to 10,000 to the inch, are not inferior. It is to be desired that transparent gratings should be obtained from first-class ruling machines. To save the diamond point it might be possible to use something softer than ordinary glass as the material of the plate.

9. _Talbot's Bands._--These very remarkable bands are seen under certain conditions when a tolerably pure spectrum is regarded with the naked eye, or with a telescope, _half the aperture being covered by a thin plate_, e.g. _of glass or mica_. The view of the matter taken by the discoverer (_Phil. Mag._, 1837, 10, p. 364) was that any ray which suffered in traversing the plate a retardation of an odd number of half wave-lengths would be extinguished, and that thus the spectrum would be seen interrupted by a number of dark bars. But this explanation cannot be accepted as it stands, being open to the same objection as Arago's theory of stellar scintillation.[9] It is as far as possible from being true that a body emitting homogeneous light would disappear on merely covering half the aperture of vision with a half-wave plate. Such a conclusion would be in the face of the principle of energy, which teaches plainly that the retardation in question leaves the aggregate brightness unaltered. The actual formation of the bands comes about in a very curious way, as is shown by a circumstance first observed by Brewster. When the retarding plate is held on the side towards the red of the spectrum, _the bands are not seen_. Even in the contrary case, the thickness of the plate must not exceed a certain limit, dependent upon the purity of the spectrum. A satisfactory explanation of these bands was first given by Airy (_Phil. Trans._, 1840, 225; 1841, 1), but we shall here follow the investigation of Sir G. G. Stokes (_Phil. Trans._, 1848, 227), limiting ourselves, however, to the case where the retarded and unretarded beams are contiguous and of equal width.

The aperture of the unretarded beam may thus be taken to be limited by x = -h, x = 0, y = -l, y= +l; and that of the beam retarded by R to be given by x = 0, x = h, y= -l, y = +l. For the former (1) § 3 gives _ _ 1 / 0 / +l / x[xi] + y[eta]\ - --------- | | sin k (at - f + -------------- )dxdy [lambda]f _/-h _/-l \ f /

2lh f k[eta]l 2f k[xi]h / [xi]h \ = - --------- · ------- sin ------- · ------ sin ------ · sin k (at - f - ----- ) (1), [lambda]f k[eta]l f k[xi]h 2f \ 2f /

on integration and reduction.

For the retarded stream the only difference is that we must subtract R from at, and that the limits of x are 0 and +h. We thus get for the disturbance at [xi], [eta], due to this stream

2lh f k[eta]l 2f k[xi]h / [xi]h \ - --------- · ------- sin ------- · ------ sin ------ . sin k (at - f - R + ----- ) (2). [lambda]f k[eta]l f k[xi]h 2f \ 2f /

If we put for shortness [pi] for the quantity under the last circular function in (1), the expressions (1), (2) may be put under the forms u sin [tau], v sin ([tau] - [alpha]) respectively; and, if I be the intensity, I will be measured by the sum of the squares of the coefficients of sin [tau] and cos [tau] in the expression

u sin[tau] + v sin([tau] - [alpha]),

so that

I = u² + v² + 2uv cos[alpha],

which becomes on putting for u, v, and [alpha] their values, and putting

/ f k[eta]l \² ( ------- sin ------- ) = Q (3), \k[eta]l f / _ _ 4l² [pi][xi]h | / 2[pi]R 2[pi][xi]h\ | I = Q · ---------- sin² --------- |2 + 2 cos ( -------- - ---------- ) | (4). [pi]²[xi]² [lambda]f |_ \[lambda] [lambda]f / _|

If the subject of examination be a luminous line parallel to [eta], we shall obtain what we require by integrating (4) with respect to [eta] from -[oo] to +[oo]. The constant multiplier is of no especial interest so that we may take as applicable to the image of a line _ _ 2 [pi][xi]h | / 2[pi]R 2[pi][xi]h \ | I = ----- sin² --------- |1 + cos ( -------- - ---------- ) | (5). [xi]² [lambda]f |_ \[lambda] [lambda]f / _|

If R = ½[lambda], I vanishes at [xi]= 0; but the whole illumination, represented by _ / +[oo] | I d[xi], is independent of the value of R. If R = 0, _/-[oo]

1 2[pi][xi]h I = ----- sin² ----------, [xi]² [lambda]f

in agreement with § 3, where a has the meaning here attached to 2h.

The expression (5) gives the illumination at [xi] due to that part of the complete image whose geometrical focus is at [xi] = 0, the retardation for this component being R. Since we have now to integrate for the whole illumination at a particular point O due to all the components which have their foci in its neighbourhood, we may conveniently regard O as origin. [xi] is then the co-ordinate relatively to O of any focal point O' for which the retardation is R; and the required result is obtained by simply integrating (5) with respect to [xi] from -[oo] to +[oo]. To each value of [xi] corresponds a different value of [lambda], and (in consequence of the dispersing power of the plate) of R. The variation of [lambda] may, however, be neglected in the integration, except in 2[pi]R/[lambda], where a small variation of [lambda] entails a comparatively large alteration of phase. If we write

[rho] = 2[pi]R/[lambda] (6),

we must regard [rho] as a function of [xi], and we may take with sufficient approximation under any ordinary circumstances

[rho] = [rho]' + [=omega][xi] (7),

where [rho]' denotes the value of [rho] at O, and [=omega] is a constant, which is positive when the retarding plate is held at the side on which the lue of the spectrum _is seen_. The possibility of dark bands depends upon [=omega] being positive. Only in this case can

cos {[rho]' + ([=omega] - 2[pi]h/[lambda]f)[xi]}

retain the constant value -1 throughout the integration, and then only when

[=omega] = 2[pi]h / [lambda]f (8)

and

cos [rho]' = -1 (9).

The first of these equations is the condition for the formation of dark bands, and the second marks their situation, which is the same as that determined by the imperfect theory.

The integration can be effected without much difficulty. For the first term in (5) the evaluation is effected at once by a known formula. In the second term if we observe that

cos {[rho]' +([=omega] - 2[pi]h/[lambda]f)[xi]} = cos {[rho]'- g1[xi]} = cos [rho]' cos g1[xi] + sin [rho]' sin g1[xi],

we see that the second part vanishes when integrated, and that the remaining integral is of the form

_+[oo] / d[xi] w = | sin² h1[xi] cos g1[xi] -----, _/-[oo] [xi]²

where

h1 = [pi]h/[lambda]f, g1 = [omega] - 2[pi]h/[lambda]f (10).

By differentiation with respect to g1 it may be proved that

w = 0 from g1 = -[oo] to g1 = -2h1, w = ½[pi](2h1 + g1) from g1 = -2h1 to g1 = 0, w = ½[pi](2h1 - g1) from g1 = 0 to g1 = 2h1, w = 0 from g1 = 2h1 to g1 = [oo].

The integrated intensity, I', or

2[pi]h1 + 2 cos[rho]w,

is thus

I' = 2[pi]h1 (11),

when g1 numerically exceeds 2h1; and, when g1 lies between ±2h1,

I = [pi]2h1 + (2h1 - [sqrt] g1²) cos[rho]' (12).

It appears therefore that there are no bands at all unless [omega] lies between 0 and +4h1, and that within these limits the best bands are formed at the middle of the range when [omega] = 2h1. The formation of bands thus requires that the retarding plate be held upon the side already specified, so that [omega] be positive; and that the thickness of the plate (to which [omega] is proportional) do not exceed a certain limit, which we may call 2T0. At the best thickness T0 the bands are black, and not otherwise.

The linear width of the band (e) is the increment of [xi] which alters [rho] by 2[pi], so that

e = 2[pi]/[=omega] (13).

With the best thickness

[=omega] = 2[pi]h/[lambda]f (14),

so that in this case

e = [lambda]f/h (15).

The bands are thus of the same width as those due to two infinitely narrow apertures coincident with the central lines of the retarded and unretarded streams, the subject of examination being itself a fine luminous line.

If it be desired to see a given number of bands in the whole or in any part of the spectrum, the thickness of the retarding plate is thereby determined, independently of all other considerations. But in order that the bands may be really visible, and still more in order that they may be black, another condition must be satisfied. It is necessary that the aperture of the pupil be accommodated to the angular extent of the spectrum, or reciprocally. Black bands will be too fine to be well seen unless the aperture (2h) of the pupil be somewhat contracted. One-twentieth to one-fiftieth of an inch is suitable. The aperture and the number of bands being both fixed, the condition of blackness determines the angular magnitude of a band and of the spectrum. The use of a grating is very convenient, for not only are there several spectra in view at the same time, but the dispersion can be varied continuously by sloping the grating. The slits may be cut out of tin-plate, and half covered by mica or "microscopic glass," held in position by a little cement.

If a telescope be employed there is a distinction to be observed, according as the half-covered aperture is between the eye and the ocular, or in front of the object-glass. In the former case the function of the telescope is simply to increase the dispersion, and the formation of the bands is of course independent of the particular manner in which the dispersion arises. If, however, the half-covered aperture be in front of the object-glass, the phenomenon is magnified as a whole, and the desirable relation between the (unmagnified) dispersion and the aperture is the same as without the telescope. There appears to be no further advantage in the use of a telescope than the increased facility of accommodation, and for this of course a very low power suffices.

The original investigation of Stokes, here briefly sketched, extends also to the case where the streams are of unequal width h, k, and are separated by an interval 2g. In the case of unequal width the bands cannot be black; but if h = k, the finiteness of 2g does not preclude the formation of black bands.

The theory of Talbot's bands with a half-covered _circular_ aperture has been considered by H. Struve (_St Peters. Trans._, 1883, 31, No. 1).

The subject of "Talbot's bands" has been treated in a very instructive manner by A. Schuster (_Phil. Mag._, 1904), whose point of view offers the great advantage of affording an instantaneous explanation of the peculiarity noticed by Brewster. A plane _pulse_, i.e. a disturbance limited to an infinitely thin slice of the medium, is supposed to fall upon a parallel grating, which again may be regarded as formed of infinitely thin wires, or infinitely narrow lines traced upon glass. The secondary pulses diverted by the ruling fall upon an object-glass as usual, and on arrival at the focus constitute a procession equally spaced in time, the interval between consecutive members depending upon the obliquity. If a retarding plate be now inserted so as to operate upon the pulses which come from one side of the grating, while leaving the remainder unaffected, we have to consider what happens at the focal point chosen. A full discussion would call for the formal application of Fourier's theorem, but some conclusions of importance are almost obvious.

Previously to the introduction of the plate we have an effect corresponding to wave-lengths closely grouped around the principal wave-length, viz. [sigma] sin [phi], where [sigma] is the grating-interval and [phi] the obliquity, the closeness of the grouping increasing with the number of intervals. In addition to these wave-lengths there are other groups centred round the wave-lengths which are submultiples of the principal one--the overlapping spectra of the second and higher orders. Suppose now that the plate is introduced so as to cover naif the aperture and that it retards those pulses which would otherwise arrive first. The consequences must depend upon the amount of the retardation. As this increases from zero, the two processions which correspond to the two halves of the aperture begin to overlap, and the overlapping gradually increases until there is almost complete superposition. The stage upon which we will fix our attention is that where the one procession bisects the intervals between the other, so that a new simple procession is constituted, containing the same number of members as before the insertion of the plate, but now spaced at intervals only half as great. It is evident that the effect at the focal point is the obliteration of the first and other spectra of odd order, so that as regards the spectrum of the first order we may consider that the two beams _interfere_. The formation of black bands is thus explained, and it requires that the plate be introduced upon one particular side, and that the amount of the retardation be adjusted to a particular value. If the retardation be too little, the overlapping of the processions is incomplete, so that besides the procession of half period there are residues of the original processions of full period. The same thing occurs if the retardation be too great. If it exceed the double of the value necessary for black bands, there is again no overlapping and consequently no interference. If the plate be introduced upon the other side, so as to retard the procession originally in arrear, there is no overlapping, whatever may be the amount of retardation. In this way the principal features of the phenomenon are accounted for, and Schuster has shown further how to extend the results to spectra having their origin in prisms instead of gratings.

10. _Diffraction when the Source of Light is not seen in Focus._--The phenomena to be considered under this head are of less importance than those investigated by Fraunhofer, and will be treated in less detail; but in view of their historical interest and of the ease with which many of the experiments may be tried, some account of their theory cannot be omitted. One or two examples have already attracted our attention when considering Fresnel's zones, viz. the shadow of a circular disk and of a screen circularly perforated.

Fresnel commenced his researches with an examination of the fringes, external and internal, which accompany the shadow of a narrow opaque strip, such as a wire. As a source of light he used sunshine passing through a very small hole perforated in a metal plate, or condensed by a lens of short focus. In the absence of a heliostat the latter was the more convenient. Following, unknown to himself, in the footsteps of Young, he deduced the principle of interference from the circumstance that the darkness of the interior bands requires the co-operation of light from both sides of the obstacle. At first, too, he followed Young in the view that the exterior bands are the result of interference between the direct light and that reflected from the edge of the obstacle, but he soon discovered that the character of the edge--e.g. whether it was the cutting edge or the back of a razor--made no material difference, and was thus led to the conclusion that the explanation of these phenomena requires nothing more than the application of Huygens's principle to the unobstructed parts of the wave. In observing the bands he received them at first upon a screen of finely ground glass, upon which a magnifying lens was focused; but it soon appeared that the ground glass could be dispensed with, the diffraction pattern being viewed in the same way as the image formed by the object-glass of a telescope is viewed through the eye-piece. This simplification was attended by a great saving of light, allowing measures to be taken such as would otherwise have presented great difficulties.

In theoretical investigations these problems are usually treated as of two dimensions only, everything being referred to the plane passing through the luminous point and perpendicular to the diffracting edges, supposed to be straight and parallel. In strictness this idea is appropriate only when the source is a luminous line, emitting cylindrical waves, such as might be obtained from a luminous point with the aid of a cylindrical lens. When, in order to apply Huygens's principle, the wave is supposed to be broken up, the phase is the same at every element of the surface of resolution which lies upon a line perpendicular to the plane of reference, and thus the effect of the whole line, or rather infinitesimal strip, is related in a constant manner to that of the element which lies in the plane of reference, and may be considered to be represented thereby. The same method of representation is applicable to spherical waves, issuing from a _point_, if the radius of curvature be large; for, although there is variation of phase along the length of the infinitesimal strip, the whole effect depends practically upon that of the central parts where the phase is sensibly constant.[10]

In fig. 17 APQ is the arc of the circle representative of the wave-front of resolution, the centre being at O, and the radius QA being equal to a. B is the point at which the effect is required, distant a + b from O, so that AB = b, AP = s, PQ = ds.

Taking as the standard phase that of the secondary wave from A, we may represent the effect of PQ by

/t [delta] \ cos 2[pi] ( - - -------- )·ds, \r [lambda]/

where [delta] = BP - AP is the retardation at B of the wave from P relatively to that from A.

Now

[delta] = (a + b) s²/2ab (1),

so that, if we write

2[pi][delta] = [pi](a + b)s² [pi]v² ------------ --------------- = ------ (2), [lambda] ab[lambda] 2

the effect at B is _ _ /ab[lambda]\½ / 2[pi]t / 2[pi]t / \ ( ---------- ) ( cos ------ | cos ½[pi]v²·dv + sin ------ | sin ½[pi]v²·dv ) (3), \2(a + b) / \ [tau] _/ [tau] _/ /

the limits of integration depending upon the disposition of the diffracting edges. When a, b, [lambda] are regarded as constant, the first factor may be omitted,--as indeed should be done for consistency's sake, inasmuch as other factors of the same nature have been omitted already.

The intensity I², the quantity with which we are principally concerned, may thus be expressed

_ _ / / \² / / \² I²= ( | cos ½[pi]v²·dv ) + ( | sin ½[pi]v²·dv ) (4). \ _/ / \ _/ /

These integrals, taken from v = 0, are known as Fresnel's integrals; we will denote them by C and S, so that _ _ / v / v C = | cos ½[pi]v²·dv, S = | cos ½[pi]v²·dv (5). _/0 _/0

When the upper limit is infinity, so that the limits correspond to the inclusion of half the primary wave, C and S are both equal to ½, by a known formula; and on account of the rapid fluctuation of sign the parts of the range beyond very moderate values of v contribute but little to the result.

Ascending series for C and S were given by K. W. Knockenhauer, and are readily investigated. Integrating by parts, we find

_v _v / i·½[pi]v² i·½[pi]v² 1 / i·½[pi]v² C + iS = | e dv = e · v - - i[pi] | e dv³; _/0 3 _/0

and, by continuing this process,

i.½[pi]v² / i[pi] i[pi] i[pi] i[pi] i[pi] i[pi] \ C + iS = e ( v - ----- v³ + ----- ----- v^5 - ----- ----- ----- v^7 + ... ). \ 3 3 5 3 5 7 /

By separation of real and imaginary parts,

C = M cos ½[pi]v² - N sin ½[pi]v² \ S = M sin ½[pi]v² - N cos ½[pi]v² / (6)

where

v [pi]²v^5 [pi]^4v^9 M = - - --------- + --------- - ... (7) 1 3·5 3·5·7·9

[pi]v³ [pi]^3v^7 [pi]^5v^11 N = ------ - --------- + ------------ ... (8) 1·3 1·3·5·7 1·3·5·7·9·11

These series are convergent for all values of v, but are practically useful only when v is small.

Expressions suitable for discussion when v is large were obtained by L. P. Gilbert (_Mem. cour. de l'Acad. de Bruxelles_, 31, p. 1). Taking

½[pi]v² = u (9),

we may write _ 1 /u e^iu du C + iS = ------------- | -------- (10). [sqrt](2[pi]) _/0 [sqrt] u

Again, by a known formula,

_[oo] 1 1 / e^-ux dx -------- = ---------- | -------- (11). [sqrt] u [sqrt][pi] _/0 [sqrt]x

Substituting this in (10), and inverting the order of integration, we get

_[oo] _u 1 / dx / e^u(i - x) C + iS = ------- | -------- | ----------- dx [sqrt]2 _/0 [sqrt] x _/0 [sqrt]x

_[oo] 1 / dx e^u(i - x) - 1 = ------- | -------- -------------- dx (12). [sqrt]2 _/0 [sqrt] x i - x

Thus, if we take

_[oo] 1 / e^-ux [sqrt](x)·dx G = ----------- | ------------------, [pi][sqrt]2 _/0 1 + x²

_[oo] 1 / e^-ux dx H = ----------- | ------------------ (13). [pi][sqrt]2 _/ [sqrt]x · (1 + x²) 0

C = ½ - G cos u + H sin u, S = ½ - G sin u - H cos u (14).

The constant parts in (14), viz. ½, may be determined by direct integration of (12), or from the observation that by their constitution G and H vanish when u = [oo], coupled with the fact that C and S then assume the value ½.

Comparing the expressions for C, S in terms of M, N, and in terms of G, H, we find that

G = ½ (cos u + sin u) - M, H = ½ (cos u - sin u) + N (15),

formulae which may be utilized for the calculation of G, H when u (or v) is small. For example, when u = 0, M = 0, N = 0, and consequently G = H = ½.

Descending series of the semi-convergent class, available for numerical calculation when u is moderately large, can be obtained from (12) by writing x = uy, and expanding the denominator in powers of y. The integration of the several terms may then be effected by the formula

_ [oo] / -y q-½ | e y dy = [Gamma](q + ½) = (q - ½)(q - 3/2) ... ½[sqrt][pi]; _/0

and we get in terms of v

1 1·3·5 1·3·5·9 G = ------- - ---------- + ----------- - (16), [pi]²v³ [pi]^4 v^7 [pi]^6 v^11

1 1·3 1·3·5·7 H = ----- - --------- + ---------- - (17). [pi]v [pi]³ v^5 [pi]^5 v^9

The corresponding values of C and S were originally derived by A. L. Cauchy, without the use of Gilbert's integrals, by direct integration by parts.

From the series for G and H just obtained it is easy to verify that

dH dG -- = - [pi]vG, -- = [pi]vH - 1 (18). dv dv

We now proceed to consider more particularly the distribution of light upon a screen PBQ near the shadow of a straight edge A. At a point P within the geometrical shadow of the obstacle, the half of the wave to the right of C (fig. 18), the nearest point on the wave-front, is wholly intercepted, and on the left the integration is to be taken from s = CA to s = [oo]. If V be the value of v corresponding to CA, viz.

/ / 2(a + b) \ V= / ( ---------- )·CA, (19), \/ \ ab[lambda] /

we may write

_[oo] _[oo] / / \² / / \² I² = ( | cos ½[pi]v²·dv ) + ( | sin ½[pi]v²·dv ) (20), \ _/v / \ _/v /

or, according to our previous notation,

I² = (½ - Cv)² + (½ - Sv)² = G² + H² (21).

Now in the integrals represented by G and H every element diminishes as V increases from zero. Hence, as CA increases, viz. as the point P is more and more deeply immersed in the shadow, the illumination _continuously_ decreases, and that without limit. It has long been known from observation that there are no bands on the interior side of the shadow of the edge.

The law of diminution when V is moderately large is easily expressed with the aid of the series (16), (17) for G, H. We have ultimately G = 0, H = ([pi]V)^-1, so that

I² = 1/[pi]²V²,

or the illumination is inversely as the square of the distance from the shadow of the edge.

For a point Q outside the shadow the integration extends over _more_ than half the primary wave. The intensity may be expressed by

I² = (½ + Cv)² + (½ + Sv)² (22);

and the maxima and minima occur when

dC dS (½ + C_v) -- + (½ + S_v) -- = 0, dV dV

whence

sin ½[pi]V² + cos ½[pi]V² = G (23).

When V = 0, viz. at the edge of the shadow, I² = ½; when V = [oo], I² = 2, on the scale adopted. The latter is the intensity due to the uninterrupted wave. The quadrupling of the intensity in passing outwards from the edge of the shadow is, however, accompanied by fluctuations giving rise to bright and dark bands. The position of these bands determined by (23) may be very simply expressed when V is large, for then sensibly G = 0, and

½[pi]V² = ¾[pi] + n[pi] (24),

n being an integer. In terms of [delta], we have from (2)

[delta] = (3/8 + ½n)[lambda] (25).

The first maximum in fact occurs when [delta] = 3/8[lambda] -.0046[lambda], and the first minimum when [delta] = 7/8[lambda] -.0016[lambda], the corrections being readily obtainable from a table of G by substitution of the approximate value of V.

The position of Q corresponding to a given value of V, that is, to a band of given order, is by (19)

a + b / / b[lambda](a + b) \ BQ = ----- AD = V / ( ----------------- ) (26). a \/ \ 2a /

By means of this expression we may trace the locus of a band of given order as b varies. With sufficient approximation we may regard BQ and b as rectangular co-ordinates of Q. Denoting them by x, y, so that AB is axis of y and a perpendicular through A the axis of x, and rationalizing (26), we have

2ax² - V²[lambda]y² - V²a[lambda]y = 0,

which represents a hyperbola with vertices at O and A.

From (24), (26) we see that the width of the bands is of the order [sqrt] {b[lambda](a + b)/a}. From this we may infer the limitation upon the width of the source of light, in order that the bands may be properly formed. If [omega] be the apparent magnitude of the source seen from A, [omega]b should be much smaller than the above quantity, or

[omega] < [sqrt] {[lambda](a + b)/ab} (27).

If a be very great in relation to b, the condition becomes

[omega] < [sqrt] ([lambda]/b) (28).

so that if b is to be moderately great (1 metre), the apparent magnitude of the sun must be greatly reduced before it can be used as a source. The values of V for the maxima and minima of intensity, and the magnitudes of the latter, were calculated by Fresnel. An extract from his results is given in the accompanying table.

+--------------------+----------+------------+ | | V | I² | +--------------------+----------+------------+ | First maximum | 1.2172 | 2.7413 | | First minimum | 1.8726 | 1.5570 | | Second maximum | 2.3449 | 2.3990 | | Second minimum | 2.7392 | 1.6867 | | Third maximum. | 3.0820 | 2.3022 | | Third minimum | 3.3913 | 1.7440 | +--------------------+----------+------------+

A very thorough investigation of this and other related questions, accompanied by fully worked-out tables of the functions concerned, will be found in a paper by E. Lommel (_Abh. bayer. Akad. d. Wiss._ II. CI., 15, Bd., iii. Abth., 1886).

When the functions C and S have once been calculated, the discussion of various diffraction problems is much facilitated by the idea, due to M. A. Cornu (_Journ. de Phys._, 1874, 3, p. 1; a similar suggestion was made independently by G. F. Fitzgerald), of exhibiting as a curve the relationship between C and S, considered as the rectangular co-ordinates (x, y) of a point. Such a curve is shown in fig. 19, where, according to the definition (5) of C, S,

_ v _ v / / x = | cos ½[pi]v²·dv, y = | sin ½[pi]v²·dv (29). _/0 _/0

The origin of co-ordinates O corresponds to v = 0; and the asymptotic points J, J', round which the curve revolves in an ever-closing spiral, correspond to v = ±[oo].

The intrinsic equation, expressing the relation between the arc [sigma] (measured from O) and the inclination [phi] of the tangent at any points to the axis of x, assumes a very simple form. For

dx = cos ½[pi]v²·dv, dy = sin ½[pi]v²·dv;

so that _ / [sigma] = | [sqrt] (dx² + dy²) = v, (30), _/

[phi] = tan^-1 (dy/dx) = ½[pi]v² (31).

Accordingly,

[phi] = ½[pi][sigma]² (32);

and for the curvature,

d[phi]/d[sigma] = [pi][sigma] (33).

Cornu remarks that this equation suffices to determine the general character of the curve. For the osculating circle at any point includes the whole of the curve which lies beyond; and the successive convolutions envelop one another without intersection.

The utility of the curve depends upon the fact that the elements of arc represent, in amplitude and phase, the component vibrations due to the corresponding portions of the primary wave-front. For by (30) d[sigma] = dv, and by (2) dv is proportional to ds. Moreover by (2) and (31) the retardation of phase of the elementary vibration from PQ (fig. 17) is 2[pi][delta]/[lambda], or [phi]. Hence, in accordance with the rule for compounding vector quantities, the resultant vibration at B, due to any finite part of the primary wave, is represented in amplitude and phase by the chord joining the extremities of the corresponding arc ([sigma]2 - [sigma]1).

In applying the curve in special cases of diffraction to exhibit the effect at any point P (fig. 18) the centre of the curve O is to be considered to correspond to that point C of the primary wave-front which lies nearest to P. The operative part, or parts, of the curve are of course those which represent the unobstructed portions of the primary wave.

Let us reconsider, following Cornu, the diffraction of a screen unlimited on one side, and on the other terminated by a straight edge. On the illuminated side, at a distance from the shadow, the vibration is represented by JJ'. The co-ordinates oí J, J' being (½, ½), (-½, -½), I² is 2; and the phase is 1/8 period in arrear of that of the element at O. As the point under contemplation is supposed to approach the shadow, the vibration is represented by the chord drawn from J to a point on the other half of the curve, which travels inwards from J' towards O. The amplitude is thus subject to fluctuations, which increase as the shadow is approached. At the point O the intensity is one-quarter of that of the entire wave, and after this point is passed, that is, when we have entered the geometrical shadow, the intensity falls off gradually to zero, _without fluctuations_. The whole progress of the phenomenon is thus exhibited to the eye in a very instructive manner.

We will next suppose that the light is transmitted by a slit, and inquire what is the effect of varying the width of the slit upon the illumination at the projection of its centre. Under these circumstances the arc to be considered is bisected at O, and its length is proportional to the width of the slit. It is easy to see that the length of the chord (which passes in all cases through O) increases to a maximum near the place where the phase-retardation is 3/8 of a period, then diminishes to a minimum when the retardation is about 7/8 of a period, and so on.

If the slit is of constant width and we require the illumination at various points on the screen behind it, we must regard the arc of the curve as of _constant length_. The intensity is then, as always, represented by the square of the length of the chord. If the slit be narrow, so that the arc is short, the intensity is constant over a wide range, and does not fall off to an important extent until the discrepancy of the extreme phases reaches about a quarter of a period.

We have hitherto supposed that the shadow of a diffracting obstacle is received upon a diffusing screen, or, which comes to nearly the same thing, is observed with an eye-piece. If the eye, provided if necessary with a perforated plate in order to reduce the aperture, be situated inside the shadow at a place where the illumination is still sensible, and be focused upon the diffracting edge, the light which it receives will appear to come from the neighbourhood of the edge, and will present the effect of a silver lining. This is doubtless the explanation of a "pretty optical phenomenon, seen in Switzerland, when the sun rises from behind distant trees standing on the summit of a mountain."[11]

II. _Dynamical Theory of Diffraction._--The explanation of diffraction phenomena given by Fresnel and his followers is independent of special views as to the nature of the aether, at least in its main features; for in the absence of a more complete foundation it is impossible to treat rigorously the mode of action of a solid obstacle such as a screen. But, without entering upon matters of this kind, we may inquire in what manner a primary wave may be resolved into elementary secondary waves, and in particular as to the law of intensity and polarization in a secondary wave as dependent upon its direction of propagation, and upon the character as regards polarization of the primary wave. This question was treated by Stokes in his "Dynamical Theory of Diffraction" (_Camb. Phil. Trans._, 1849) on the basis of the elastic solid theory.

Let x, y, z be the co-ordinates of any particle of the medium in its natural state, and [chi], [eta], [zeta] the displacements of the same particle at the end of time t, measured in the directions of the three axes respectively. Then the first of the equations of motion may be put under the form

d²[xi] /d²[xi] d²[xi] d²[xi]\ d² /d²[xi] d²[eta] d²[zeta]\ ------ = b²( ------ + ------ + ------ ) + (a² - b²)--( ------ + ------- + -------- ), dt² \ dx² dy² dz² / dx \ dx² dy² dz² /

where a2 and b2 denote the two arbitrary constants. Put for shortness

d²[xi] d²[eta] d²[zeta] ------ + ------- + -------- = [delta] (1), dx² dy² dz²

and represent by [Delta]²[chi] the quantity multiplied by b². According to this notation, the three equations of motion are

d²[xi] d[delta] \ ------ = b²[Delta]²[xi] + (a² - b²) -------- | dt² dx | | d²[eta] d[delta] | ------- = b²[Delta]²[eta] + (a² - b²) -------- > (2). dt² dy | | d²[zeta] d[delta] | -------- = b²[Delta]²[zeta] + (a² - b²) -------- | dt² dz /

It is to be observed that S denotes the dilatation of volume of the element situated at (x, y, z). In the limiting case in which the medium is regarded as absolutely incompressible [delta] vanishes; but, in order that equations (2) may preserve their generality, we must suppose a at the same time to become infinite, and replace a²[delta] by a new function of the co-ordinates.

These equations simplify very much in their application to plane waves. If the ray be parallel to OX, and the direction of vibration parallel to OZ, we have [xi] = 0, [eta] = 0, while [zeta] is a function of x and t only. Equation (1) and the first pair of equations (2) are thus satisfied identically. The third equation gives

d²[zeta] d²[zeta] -------- = -------- (3), dt² dx²

of which the solution is

[zeta] = f(bt - x) (4),

where f is an arbitrary function.

The question as to the law of the secondary waves is thus answered by Stokes. "Let [xi] = 0, [eta] = 0, [zeta] = f(bt-x) be the displacements corresponding to the incident light; let O1 be any point in the plane P (of the wave-front), dS an element of that plane adjacent to O1, and consider the disturbance due to that portion only of the incident disturbance which passes continually across dS. Let O be any point in the medium situated at a distance from the point O1 which is large in comparison with the length of a wave; let O1O = r, and let this line make an angle [theta] with the direction of propagation of the incident light, or the axis of x, and [phi] with the direction of vibration, or axis of z. Then the displacement at O will take place in a direction perpendicular to O1O, and lying in the plane ZO1O; and, if [zeta]' be the displacement at O, reckoned positive in the direction nearest to that in which the incident vibrations are reckoned positive,

dS [zeta]' = ------ ( 1 + cos[theta]) sin[phi] f'(bt - r). 4[pi]r

In particular, if

2[pi] f(bt - x) = c sin -------- (bt - x) (5), [lambda]

we shall have

cdS 2[pi] [zeta]' = ---------- (1 + cos[theta]) sin[phi]cos -------- (bt - r) (6)." 2[lambda]r [lambda]

It is then verified that, after integration with respect to dS, (6) gives the same disturbance as if the primary wave had been supposed to pass on unbroken.

The occurrence of sin [phi] as a factor in (6) shows that the relative intensities of the primary light and of that diffracted in the direction [theta] depend upon the condition of the former as regards polarization. If the direction of primary vibration be perpendicular to the plane of diffraction (containing both primary and secondary rays), sin [phi] = 1; but, if the primary vibration be in the plane of diffraction, sin [phi] = cos [theta]. This result was employed by Stokes as a criterion of the direction of vibration; and his experiments, conducted with gratings, led him to the conclusion that the vibrations of polarized light are executed in a direction _perpendicular_ to the plane of polarization.

The factor (1 + cos [theta]) shows in what manner the secondary disturbance depends upon the direction in which it is propagated with respect to the front of the primary wave.

If, as suffices for all practical purposes, we limit the application of the formulae to points in advance of the plane at which the wave is supposed to be broken up, we may use simpler methods of resolution than that above considered. It appears indeed that the purely mathematical question has no definite answer. In illustration of this the analogous problem for sound may be referred to. Imagine a flexible lamina to be introduced so as to coincide with the plane at which resolution is to be effected. The introduction of the lamina (supposed to be devoid of inertia) will make no difference to the propagation of plane parallel sonorous waves through the position which it occupies. At every point the motion of the lamina will be the same as would have occurred in its absence, the pressure of the waves impinging from behind being just what is required to generate the waves in front. Now it is evident that the aerial motion in front of the lamina is determined by what happens at the lamina without regard to the cause of the motion there existing. Whether the necessary forces are due to aerial pressures acting on the rear, or to forces directly impressed from without, is a matter of indifference. The conception of the lamina leads immediately to two schemes, according to which a primary wave may be supposed to be broken up. In the first of these the element dS, the effect of which is to be estimated, is supposed to execute its actual motion, while every other element of the plane lamina is maintained at rest. The resulting aerial motion in front is readily calculated (see Rayleigh, _Theory of Sound_, § 278); it is symmetrical with respect to the origin, i.e. independent of [theta]. When the secondary disturbance thus obtained is integrated with respect to dS over the entire plane of the lamina, the result is necessarily the same as would have been obtained had the primary wave been supposed to pass on without resolution, for this is precisely the motion generated when every element of the lamina vibrates with a common motion, equal to that attributed to dS. The only assumption here involved is the evidently legitimate one that, when two systems of variously distributed motion at the lamina are superposed, the corresponding motions in front are superposed also.

The method of resolution just described is the simplest, but it is only one of an indefinite number that might be proposed, and which are all equally legitimate, so long as the question is regarded as a merely mathematical one, without reference to the physical properties of actual screens. If, instead of supposing the _motion_ at dS to be that of the primary wave, and to be zero elsewhere, we suppose the _force_ operative over the element dS of the lamina to be that corresponding to the primary wave, and to vanish elsewhere, we obtain a secondary wave following quite a different law. In this case the motion in different directions varies as cos[theta], vanishing at right angles to the direction of propagation of the primary wave. Here again, on integration over the entire lamina, the aggregate effect of the secondary waves is necessarily the same as that of the primary.

In order to apply these ideas to the investigation of the secondary wave of light, we require the solution of a problem, first treated by Stokes, viz. the determination of the motion in an infinitely extended elastic solid due to a locally applied periodic force. If we suppose that the force impressed upon the element of mass D dx dy dz is

DZ dx dy dz,

being everywhere parallel to the axis of Z, the only change required in our equations (1), (2) is the addition of the term Z to the second member of the third equation (2). In the forced vibration, now under consideration, Z, and the quantities [xi], [eta], [zeta], [delta] expressing the resulting motion, are to be supposed proportional to e^int, where i = [sqrt](-1), and n = 2[pi]/[tau], [tau] being the periodic time. Under these circumstances the double differentiation with respect to t of any quantity is equivalent to multiplication by the factor -n², and thus our equations take the form

d[delta] \ (b²[Delta]² + n²)[xi] + (a² - b²) -------- = 0 | dx | | d[delta] | (b²[Delta]² + n²)[eta] + (a² - b²) -------- = 0 > (7). dx | | d[delta] | (b²[Delta]² + n²)[zeta] + (a² - b²) -------- = -Z | dx /

It will now be convenient to introduce the quantities.[=omega]1, [=omega]2, [=omega]3 which express the _rotations_ of the elements of the medium round axes parallel to those of co-ordinates, in accordance with the equations

d[xi] d[eta] d[eta] d[zeta] [=omega]3 = ----- - ------, [=omega]1 = ------ - -------, dy dx' dz dy

d[zeta] d[xi] [=omega]2 = ------- - ----- (8). dx dz

In terms of these we obtain from (7), by differentiation and subtraction,

(b²[Delta]² + n²) [=omega]3 = 0 \ (b²[Delta]² + n²) [=omega]1 = dZ/dy > (9). (b²[Delta]² + n²) [=omega]2 = -dZ/dx /

The first of equations (9) gives

[=omega]3 = 0 (10).

For =[omega]1, we have _ _ _ -ikr 1 / / / dZ e [=omega]1 = ------- | | | -- ----- dx dy dz (11), 4[pi]b² _/_/_/ dy r

where r is the distance between the element dx dy dz and the point where [=omega]1 is estimated, and

k = n/b = 2[pi]/[lambda] (12),

[lambda] being the wave-length.

(This solution may be verified in the same manner as Poisson's theorem, in which k = 0.)

We will now introduce the supposition that the force Z acts only within a small space of volume T, situated at (x, y, z), and for simplicity suppose that it is at the origin of co-ordinates that the rotations are to be estimated. Integrating by parts in (11), we get

_ -ikr _ _ _ / e dZ | Ze^-ikr | / d / e^-ikr\ | ------ -- dy = | ------- | - | Z -- ( ------- ) dy, _/ r dy |_ r _| _/ dy \ r /

in which the integrated terms at the limits vanish, Z being finite only within the region T. Thus

_ _ _ -ikr 1 / / / d /e^ \ [=omega]1 = ------- | | | Z -- ( -------- ) dx dy dz. 4[pi]b² _/_/_/ dy \ r /

Since the dimensions of T are supposed to be very small in comparison with [lambda], the factor d/dy (e^-ikr / r) is sensibly constant; so that, if Z stand for the mean value of Z over the volume T, we may write

TZ y d / e^-ikr \ [=omega]1 = ------- · - · -- ( ------ ) (13). 4[pi]b² r dr \ r /

In like manner we find

TZ x d / e^-ikr \ [=omega]2 = ------ · - · -- ( ------- ) (14). 4[pi]b² r dr \ r /

From (10), (13), (14) we see that, as might have been expected, the rotation at any point is about an axis perpendicular both to the direction of the force and to the line joining the point to the source of disturbance. If the resultant rotation be [omega], we have

TZ [sqrt](x² + y²) d /e^-ikr\ [=omega] = ------- · --------------- · -- ( ------ ) = 4[pi]b² r dr \ r /

TZ sin[phi] d /e^-ikr\ = ----------- -- ( ------ ), 4[pi]b² dr \ r /

[phi] denoting the angle between r and z. In differentiating e^(-ikr)/r with respect to r, we may neglect the term divided by r² as altogether insensible, kr being an exceedingly great quantity at any moderate distance from the origin of disturbance. Thus

-ik·TZ sin[phi] /e^-ikr\ [=omega] = --------------- · ( ------ ) (15), 4[pi]b² \ r /

which completely determines the rotation at any point. For a disturbing force of given integral magnitude it is seen to be everywhere about an axis perpendicular to r and the direction of the force, and in magnitude dependent only upon the angle ([phi]) between these two directions and upon the distance (r).

The intensity of light is, however, more usually expressed in terms of the actual displacement in the plane of the wave. This displacement, which we may denote by [zeta]', is in the plane containing z and r, and perpendicular to the latter. Its connexion with [=omega]is expressed by [=omega] = d[zeta]'/dr; so that

TZ sin [phi] /e^-ikr\ [zeta]' = ----------- · ( ------ ) (16), 4[pi]b² \ r /

where the factor e^int is restored.

Retaining only the real part of (16), we find, as the result of a local application of force equal to

DTZ cos nt (17),

the disturbance expressed by

TZ sin [phi] /cos(nt - kr)\ [zeta]' = ------------ · ( ------------ ) (18). 4[pi]b² \ r /

The occurrence of sin [phi] shows that there is no disturbance radiated in the direction of the force, a feature which might have been anticipated from considerations of symmetry.

We will now apply (18) to the investigation of a law of secondary disturbance, when a primary wave

[zeta] = sin(nt - kx) (19)

is supposed to be broken up in passing the plane x = 0. The first step is to calculate the force which represents the reaction between the parts of the medium separated by x = 0. The force operative upon the positive half is parallel to OZ, and of amount per unit of area equal to

-b²D d[zeta]/dx = b²kD cos nt;

and to this force acting over the whole of the plane the actual motion on the positive side may be conceived to be due. The secondary disturbance corresponding to the element dS of the plane may be supposed to be that caused by a force of the above magnitude acting over dS and vanishing elsewhere; and it only remains to examine what the result of such a force would be.

Now it is evident that the force in question, supposed to act upon the positive half only of the medium, produces just double of the effect that would be caused by the same force if the medium were undivided, and on the latter supposition (being also localized at a point) it comes under the head already considered. According to (18), the effect of the force acting at dS parallel to OZ, and of amount equal to

2b²kD dS cos nt,

will be a disturbance

dS sin [phi] [zeta]' = ------------ cos(nt - kr) (20), [lambda]r

regard being had to (12). This therefore expresses the secondary disturbance at a distance r and in a direction making an angle [phi] with OZ (the direction of primary vibration) due to the element dS of the wave-front.

The proportionality of the secondary disturbance to sin [phi] is common to the present law and to that given by Stokes, but here there is no dependence upon the angle [theta] between the primary and secondary rays. The occurrence of the factor [lambda]r^-1, and the necessity of supposing the phase of the secondary wave accelerated by a quarter of an undulation, were first established by Archibald Smith, as the result of a comparison between the primary wave, supposed to pass on without resolution, and the integrated effect of all the secondary waves (§ 2). The occurrence of factors such as sin [phi], or ½(1 + cos [theta]), in the expression of the secondary wave has no influence upon the result of the integration, the effects of all the elements for which the factors differ appreciably from unity being destroyed by mutual interference.

The choice between various methods of resolution, all mathematically admissible, would be guided by physical considerations respecting the mode of action of obstacles. Thus, to refer again to the acoustical analogue in which plane waves are incident upon a perforated rigid screen, the circumstances of the case are best represented by the first method of resolution, leading to symmetrical secondary waves, in which the normal motion is supposed to be zero over the unperforated parts. Indeed, if the aperture is very small, this method gives the correct result, save as to a constant factor. In like manner our present law (20) would apply to the kind of obstruction that would be caused by an actual physical division of the elastic medium, extending over the whole of the area supposed to be occupied by the intercepting screen, but of course not extending to the parts supposed to be perforated.

On the electromagnetic theory, the problem of diffraction becomes definite when the properties of the obstacle are laid down. The simplest supposition is that the material composing the obstacle is perfectly conducting, i.e. perfectly reflecting. On this basis A. J. W. Sommerfeld (_Math. Ann._, 1895, 47, p. 317), with great mathematical skill, has solved the problem of the shadow thrown by a semi-infinite plane screen. A simplified exposition has been given by Horace Lamb (_Proc. Lond. Math. Soc._, 1906, 4, p. 190). It appears that Fresnel's results, although based on an imperfect theory, require only insignificant corrections. Problems not limited to two dimensions, such for example as the shadow of a circular disk, present great difficulties, and have not hitherto been treated by a rigorous method; but there is no reason to suppose that Fresnel's results would be departed from materially. (R.)

FOOTNOTES:

[1] The descending series for J0(z) appears to have been first given by Sir W. Hamilton in a memoir on "Fluctuating Functions," _Roy. Irish Trans._, 1840.

[2] Airy, loc. cit. "Thus the magnitude of the central spot is diminished, and the brightness of the rings increased, by covering the central parts of the object-glass."

[3] _"Man kann daraus schliessen, was moglicher Weise durch Mikroskope noch zu sehen ist. Ein mikroskopischer Gegenstand z. B, dessen Durchmesser = ([lambda]) ist, und der aus zwei Theilen besteht, kann nicht mehr als aus zwei Theilen bestehend erkannt werden. Dieses zeigt uns eine Grenze des Sehvermogens durch Mikroskope"_ (_Gilbert's Ann._ 74, 337). Lord Rayleigh has recorded that he was himself convinced by Fraunhofer's reasoning at a date antecedent to the writings of Helmholtz and Abbe.

[4] The last sentence is repeated from the writer's article "Wave Theory" in the 9th edition of this work, but A. A. Michelson's ingenious échelon grating constitutes a realization in an unexpected manner of what was thought to be impracticable.--[R.]

[5] Compare also F. F. Lippich, _Pogg. Ann._ cxxxix. p. 465, 1870; Rayleigh, _Nature_ (October 2, 1873).

[6] The power of a grating to construct light of nearly definite wave-length is well illustrated by Young's comparison with the production of a musical note by reflection of a sudden sound from a row of palings. The objection raised by Herschel (_Light_, § 703) to this comparison depends on a misconception.

[7] It must not be supposed that errors of this order of magnitude are unobjectionable in all cases. The position of the middle of the bright band representative of a mathematical line can be fixed with a spider-line micrometer within a small fraction of the width of the band, just as the accuracy of astronomical observations far transcends the separating power of the instrument.

[8] "In the same way we may conclude that in flat gratings any departure from a straight line has the effect of causing the dust in the slit and the spectrum to have different foci--a fact sometimes observed." (Rowland, "On Concave Gratings for Optical Purposes," _Phil. Mag._, September 1883).

[9] On account of inequalities in the atmosphere giving a variable refraction, the light from a star would be irregularly distributed over a screen. The experiment is easily made on a laboratory scale, with a small source of light, the rays from which, in their course towards a rather distant screen, are disturbed by the neighbourhood of a heated body. At a moment when the eye, or object-glass of a telescope, occupies a dark position, the star vanishes. A fraction of a second later the aperture occupies a bright place, and the star reappears. According to this view the chromatic effects depend entirely upon atmospheric dispersion.

[10] In experiment a line of light is sometimes substituted for a point in order to increase the illumination. The various parts of the line are here _independent_ sources, and should be treated accordingly. To assume a cylindrical form of primary wave would be justifiable only when there is synchronism among the secondary waves issuing from the various centres.

[11] H. Necker (_Phil. Mag._, November 1832); Fox Talbot (_Phil. Mag._, June 1833). "When the sun is about to emerge ... every branch and leaf is lighted up with a silvery lustre of indescribable beauty.... The birds, as Mr Necker very truly describes, appear like flying brilliant sparks." Talbot ascribes the appearance to diffraction; and he recommends the use of a telescope.

DIFFUSION (from the Lat. _diffundere; dis-_, asunder, and _fundere_, to pour out), in general, a spreading out, scattering or circulation; in physics the term is applied to a special phenomenon, treated below.

1. _General Description._--When two different substances are placed in contact with each other they sometimes remain separate, but in many cases a gradual mixing takes place. In the case where both the substances are gases the process of mixing continues until the result is a uniform mixture. In other cases the proportions in which two different substances can mix lie between certain fixed limits, but the mixture is distinguished from a chemical compound by the fact that between these limits the composition of the mixture is capable of continuous variation, while in chemical compounds, the proportions of the different constituents can only have a discrete series of numerical values, each different ratio representing a different compound. If we take, for example, air and water in the presence of each other, air will become dissolved in the water, and water will evaporate into the air, and the proportions of either constituent absorbed by the other will vary continuously. But a limit will come when the air will absorb no more water, and the water will absorb no more air, and throughout the change a definite surface of separation will exist between the liquid and the gaseous parts. When no surface of separation ever exists between two substances they must necessarily be capable of mixing in all proportions. If they are not capable of mixing in all proportions a discontinuous change must occur somewhere between the regions where the substances are still unmixed, thus giving rise to a surface of separation.

The phenomena of mixing thus involves the following processes:--(1) A motion of the substances relative to one another throughout a definite _region_ of space in which mixing is taking place. This relative motion is called "diffusion." (2) The passage of portions of the mixing substances across the _surface_ of separation when such a surface exists. These surface actions are described under various terms such as solution, evaporation, condensation and so forth. For example, when a soluble salt is placed in a liquid, the process which occurs at the surface of the salt is called "solution," but the salt which enters the liquid by solution is transported from the surface into the interior of the liquid by "diffusion."

Diffusion may take place in solids, that is, in regions occupied by matter which continues to exhibit the properties of the solid state. Thus if two liquids which can mix are separated by a membrane or partition, the mixing may take place through the membrane. If a solution of salt is separated from pure water by a sheet of parchment, part of the salt will pass through the parchment into the water. If water and glycerin are separated in this way most of the water will pass into the glycerin and a little glycerin will pass through in the opposite direction, a property frequently used by microscopists for the purpose of gradually transferring minute algae from water into glycerin. A still more interesting series of examples is afforded by the passage of gases through partitions of metal, notably the passage of hydrogen through platinum and palladium at high temperatures. When the process is considered with reference to a membrane or partition taken as a whole, the passage of a substance from one side to the other is commonly known as "osmosis" or "transpiration" (see SOLUTION), but what occurs in the material of the membrane itself is correctly described as diffusion.

Simple cases of diffusion are easily observed qualitatively. If a solution of a coloured salt is carefully introduced by a funnel into the bottom of a jar containing water, the two portions will at first be fairly well defined, but if the mixture can exist in all proportions, the surface of separation will gradually disappear; and the rise of the colour into the upper part and its gradual weakening in the lower part, may be watched for days, weeks or even longer intervals. The diffusion of a strong aniline colouring matter into the interior of gelatine is easily observed, and is commonly seen in copying apparatus. Diffusion of gases may be shown to exist by taking glass jars containing vapours of hydrochloric acid and ammonia, and placing them in communication with the heavier gas downmost. The precipitation of ammonium chloride shows that diffusion exists, though the chemical action prevents this example from forming a typical case of diffusion. Again, when a film of Canada balsam is enclosed between glass plates, the disappearance during a few weeks of small air bubbles enclosed in the balsam can be watched under the microscope.

In fluid media, whether liquids or gases, the process of mixing is greatly accelerated by stirring or agitating the fluids, and liquids which might take years to mix if left to themselves can thus be mixed in a few seconds. It is necessary to carefully distinguish the effects of agitation from those of diffusion proper. By shaking up two liquids which do not mix we split them up into a large number of different portions, and so greatly increase the area of the surface of separation, besides decreasing the thicknesses of the various portions. But even when we produce the appearance of a uniform turbid mixture, the small portions remain quite distinct. If however the fluids can really mix, the final process must in every case depend on diffusion, and all we do by shaking is to increase the sectional area, and decrease the thickness of the diffusing portions, thus rendering the completion of the operation more rapid. If a gas is shaken up in a liquid the process of absorption of the bubbles is also accelerated by capillary action, as occurs in an ordinary sparklet bottle. To state the matter precisely, however finely two fluids have been subdivided by agitation, the molecular constitution of the different portions remains unchanged. The ultimate process by which the individual molecules of two different substances become mixed, producing finally a homogeneous mixture, is in every case diffusion. In other words, diffusion is that relative motion of the molecules of two different substances by which the proportions of the molecules in any region containing a finite number of molecules are changed.

In order, therefore, to make accurate observations of diffusion in fluids it is necessary to guard against any cause which may set up currents; and in some cases this is exceedingly difficult. Thus, if gas is absorbed at the upper surface of a liquid, and if the gaseous solution is heavier than the pure liquid, currents may be set up, and a steady state of diffusion may cease to exist. This has been tested experimentally by C. G. von Hüfner and W. E. Adney. The same thing may happen when a gas is evolved into a liquid at the surface of a solid even if no bubbles are formed; thus if pieces of aluminium are placed in caustic soda, the currents set up by the evolution of hydrogen are sufficient to set the aluminium pieces in motion, and it is probable that the motions of the Diatomaceae are similarly caused by the evolution of oxygen. In some pairs of substances diffusion may take place more rapidly than in others. Of course the progress of events in any experiment necessarily depends on various causes, such as the size of the containing vessels, but it is easy to see that when experiments with different substances are carried out under similar conditions, however these "similar conditions" be defined, the rates of diffusion must be capable of numerical comparison, and the results must be expressible in terms of at least one physical quantity, which for any two substances can be called their coefficient of diffusion. How to select this quantity we shall see later.

2 _Quantitative Methods of observing Diffusion._--The simplest plan of determining the progress of diffusion between two liquids would be to draw off and examine portions from different strata at some stage in the process; the disturbance produced would, however, interfere with the subsequent process of diffusion, and the observations could not be continued. By placing in the liquid column hollow glass beads of different average densities, and observing at what height they remain suspended, it is possible to trace the variations of density of the liquid column at different depths, and different times. In this method, which was originally introduced by Lord Kelvin, difficulties were caused by the adherence of small air bubbles to the beads.

In general, optical methods are the most capable of giving exact results, and the following may be distinguished, (a) _By refraction in a horizontal plane._ If the containing vessel is in the form of a prism, the deviation of a horizontal ray of light in passing through the prism determines the index of refraction, and consequently the density of the stratum through which the ray passes, (b) _By refraction in a vertical plane._ Owing to the density varying with the depth, a horizontal ray entering the liquid also undergoes a small vertical deviation, being bent downwards towards the layers of greater density. The observation of this vertical deviation determines not the actual density, but its rate of variation with the depth, i.e. the "density gradient" at any point, (c) _By the saccharimeter._ In the cases of solutions of sugar, which cause rotation of the plane of polarized light, the density of the sugar at any depth may be determined by observing the corresponding angle of rotation, this was done originally by W. Voigt.

3. _Elementary Definitions of Coefficient of Diffusion._--The simplest case of diffusion is that of a substance, say a gas, diffusing in the interior of a homogeneous solid medium, which remains at rest, when no external forces act on the system. We may regard it as the result of experience that: (1) if the density of the diffusing substance is everywhere the same no diffusion takes place, and (2) if the density of the diffusing substance is different at different points, diffusion will take place from places of greater to those of lesser density, and will not cease until the density is everywhere the same. It follows that the rate of flow of the diffusing substance at any point in any direction must depend on the density gradient at that point in that direction, i.e. on the rate at which the density of the diffusing substance decreases as we move in that direction. We may define the _coefficient of diffusion_ as the ratio of the total mass per unit area which flows across any small section, to the rate of decrease of the density per unit distance in a direction perpendicular to that section.

In the case of steady diffusion parallel to the axis of x, if [rho] be the density of the diffusing substance, and q the mass which flows across a unit of area in a plane perpendicular to the axis of x, then the density gradient is -d[rho]/dx and the ratio of q to this is called the "coefficient of diffusion." By what has been said this ratio remains finite, however small the actual gradient and flow may be., and it is natural to assume, at any rate as a first approximation, that it is constant as far as the quantities in question are concerned. Thus if the coefficient of diffusion be denoted by K we have q= -K(d[rho]/dx).

Further, the rate at which the quantity of substance is increasing in an element between the distances x and x+dx is equal to the difference of the rates of flow in and out of the two faces, whence as in hydrodynamics, we have d[rho]/dt =-dq/dx.

It follows that the equation of diffusion in this case assumes the form

d[rho] d / d[rho] \ ------ = -- ( K ------ ), dt dx \ dx /

which is identical with the equations representing conduction of heat, flow of electricity and other physical phenomena. For motion in three dimensions we have in like manner

d[rho] d / d[rho]\ d / d[rho]\ d / d[rho]\ ------ = -- ( K ------ ) + -- ( K ------ ) + -- ( K ------ ); dt dx \ dx / dy \ dy / dz \ dz /

and the corresponding equations in electricity and heat for anisotropic substances would be available to account for any parallel phenomena, which may arise, or might be conceived, to exist in connexion with diffusion through a crystalline solid.

In the case of a very dilute solution, the coefficient of diffusion of the dissolved substance can be defined in the same way as when the diffusion takes place in a solid, because the effects of diffusion will not have any perceptible influence on the solvent, and the latter may therefore be regarded as remaining practically at rest. But in most cases of diffusion between two fluids, both of the fluids are in motion, and hence there is far greater difficulty in determining the motion, and even in defining the coefficient of diffusion. It is important to notice in the first instance, that it is only the relative motion of the two substances which constitutes diffusion. Thus when a current of air is blowing, under ordinary circumstances the changes which take place are purely mechanical, and do not depend on the separate diffusions of the oxygen and nitrogen of which the air is mainly composed. It is only when two gases are flowing with unequal velocity, that is, when they have a relative motion, that these changes of relative distribution, which are called diffusion, take place. The best way out of the difficulty is to investigate the separate motions of the two fluids, taking account of the mechanical actions exerted on them, and supposing that the mutual action of the fluids causes either fluid to resist the relative motion of the other.

4. _The Coefficient of Resistance._--Let us call the two diffusing fluids A and B. If B were absent, the motion of the fluid A would be determined entirely by the variations of pressure of the fluid A, and by the external forces, such as that due to gravity acting on A. Similarly if A were absent, the motion of B would be determined entirely by the variations of pressure due to the fluid B, and by the external forces acting on B. When both fluids are mixed together, each fluid tends to resist the relative motion of the other, and by the law of equality of action and reaction, the resistance which A experiences from B is everywhere equal and opposite to the resistance which B experiences from A. If the amount of this resistance per unit volume be divided by the relative velocity of the two fluids, and also by the product of their densities, the quotient is called the "coefficient of resistance." If then [rho]1, [rho]2 are the densities cf the two fluids, u1, u2 their velocities, C the coefficient of resistance, then the portion of the fluid A contained in a small element of volume v will experience from the fluid B a resistance C[rho]1[rho]2v(u1- u2), and the fluid B contained in the same volume element will experience from the fluid A an equal and opposite resistance, C[rho]1[rho]2v(u2 - u1).

This definition implies the following laws of resistance to diffusion, which must be regarded as based on experience, and not as self-evident truths: (1) each fluid tends to assume, so far as diffusion is concerned, the same equüibrium distribution that it would assume if its motion were unresisted by the presence of the other fluid. (Of course, the mutual attraction of gravitation of the two fluids might affect the final distribution, but this is practically negligible. Leaving such actions as this out of account the following statement is correct.) In a state of equilibrium, the density of each fluid at any point thus depends only on the partial pressure of that fluid alone, and is the same as if the other fluids were absent. It does not depend on the partial pressures of the other fluids. If this were not the case, the resistance to diffusion would be analogous to friction, and would contain terms which were independent of the relative velocity u2 - u1. (2) For slow motions the resistance to diffusion is (approximately at any rate) proportional to the relative velocity. (3) The coefficient of resistance C is not necessarily always constant; it may, for example, and, in general, does, depend on the temperature.

If we form the equations of hydrodynamics for the different fluids occurring in any mixture, taking account of diffusion, but neglecting viscosity, and using suffixes 1, 2 to denote the separate fluids, these assume the form given by James Clerk Maxwell ("Diffusion," in _Ency. Brit._, 9th ed.):--

Du1 dp1 [rho] --- + --- - X1[rho]1 + C12[rho]1[rho]2(u1 - u2) + &c. = 0, Dt dx

where

Du1 du1 du1 du1 du1 --- = --- + u1 --- + v1 --- + w1 ---, Dt dt dx dy dz

and these equations imply that when diffusion and other motions cease, the fluids satisfy the separate conditions of equilibrium dp1/dx - X1[rho]1 = 0. The assumption made in the following account is that terms such as Du1/Dt may be neglected in the cases considered.

A further property based on experience is that the motions set up in a mixture by diffusion are very slow compared with those set up by mechanical actions, such as differences of pressure. Thus, if two gases at equal temperature and pressure be allowed to mix by diffusion, the heavier gas being below the lighter, the process will take a long time; on the other hand, if two gases, or parts of the same gas, at different pressures be connected, equalization of pressure will take place almost immediately. It follows from this property that the forces required to overcome the "inertia" of the fluids in the motions due to diffusion are quite imperceptible. At any stage of the process, therefore, any one of the diffusing fluids may be regarded as in equilibrium under the action of its own partial pressure, the external forces to which it is subjected and the resistance to diffusion of the other fluids.

5. _Slow Diffusion of two Gases. Relation between the Coefficients of Resistance and of Diffusion._--We now suppose the diffusing substances to be two gases which obey Boyle's law, and that diffusion takes place in a closed cylinder or tube of unit sectional area at constant temperature, the surfaces of equal density being perpendicular to the axis of the cylinder, so that the direction of diffusion is along the length of the cylinder, and we suppose no external forces, such as gravity, to act on the system.

The densities of the gases are denoted by [rho]1, [rho]2, their velocities of diffusion by u1, u2, and if their partial pressures are p1, p2, we have by Boyle's law p1 = k1[rho]1, p2 = k2[rho]2, where k1, k2 are constants for the two gases, the temperature being constant. The axis of the cylinder is taken as the axis of x.

From the considerations of the preceding section, the effects of inertia of the diffusing gases may be neglected, and at any instant of the process either of the gases is to be treated as kept in equilibrium by its partial pressure and the resistance to diffusion produced by the other gas. Calling this resistance per unit volume R, and putting R = C[rho]1[rho]2(u1 - u2), where C is the coefficient of resistance, the equations of equilibrium give

dp1 dp2 --- + C[rho]1[rho]2(u1 - u2)= 0, and --- + C[rho]1[rho]2(u2 - u1)= 0 (1). dx dx

These involve

dp1 dp2 --- + --- = 0 or p1 + p2 = P (2) dx dx

where P is the total pressure of the mixture, and is everywhere constant, consistently with the conditions of mechanical equilibrium.

Now dp1/dx is the pressure-gradient of the first gas, and is, by Boyle's law, equal to k1 times the corresponding density-gradient. Again [rho]1u1 is the mass of gas flowing across any section per unit time, and k1[rho]1u1 or p1u1 can be regarded as representing the flux of partial pressure produced by the motion of the gas. Since the total pressure is everywhere constant, and the ends of the cylinder are supposed fixed, the fluxes of partial pressure due to the two gases are equal and opposite, so that

p1u1 + p2u2 = 0 or k1[rho]1u1 + k2[rho]2u2 = 0 (3).

From (2) (3) we find by elementary algebra

u1/p2 = - u2/p1 = (u1 - u2)/(p1 + p2) = (u1 - u2)/P,

and therefore

p2u1 = - p2u2 = p1p2(u1 - u2)/P = k1k2[rho]1[rho]2(u1 - u2)/P

Hence equations (1) (2) gives

dp1 CP dp2 CP --- + ---- (p1u1) = 0, and --- + ---- (p2u2) = 0; dx k1k2 dx k1k2

whence also substituting p1 = k1[rho]1, p2 = k2[rho]2, and by transposing

k1k2 d[rho]1 k1k2 d[rho]2 [rho]1u1 = - ---- -------, and [rho]2u2 = - ---- -------. CP dx CP dx

We may now define the "coefficient of diffusion" of either gas as the ratio of the rate of flow of that gas to its density-gradient. With this definition, the coefficients of diffusion of both the gases in a mixture are equal, each being equal to k1k2/CP. The ratios of the fluxes of partial pressure to the corresponding pressure-gradients are also equal to the same coefficient. Calling this coefficient K, we also observe that the equations of continuity for the two gases are

d[rho]1 d([rho]1u1) d[rho]2 d([rho]2u2) ------- + ----------- = 0, and ------- + ----------- = 0, dt dx dt dx

leading to the equations of diffusion

d[rho]1 d / d[rho]1\ d[rho]2 d / d[rho]2\ ------- = -- ( K ------- ) , and ------- = -- ( K ------- ), dt dx \ dx / dt dx \ dx /

exactly as in the case of diffusion through a solid.

If we attempt to treat diffusion in liquids by a similar method, it is, in the first place, necessary to define the "partial pressure" of the components occurring in a liquid mixture. This leads to the conception of "osmotic pressure," which is dealt with in the article SOLUTION. For dilute solutions at constant temperature, the assumption that the osmotic pressure is proportional to the density, leads to results agreeing fairly closely with experience, and this fact may be represented by the statement that a substance occurring in a dilute solution behaves like a perfect gas.

6. _Relation of the Coefficient of Diffusion to the Units of Length and Time._--We may write the equation defining K in the form

I d[rho] u = -K × ----- ------. [rho] dx

Here -d[rho]/[rho]dx represents the "percentage rate" at which the density decreases with the distance x; and we thus see that the coefficient of diffusion represents the ratio of the velocity of flow to the percentage rate at which the density decreases with the distance measured in the direction of flow. This percentage rate being of the nature of a number divided by a length, and the velocity being of the nature of a length divided by a time, we may state that K is of two dimensions in length and - 1 in time, i.e. dimensions L²/T.

_Example 1._ Taking K = 0.1423 for carbon dioxide and air (at temperature 0° C. and pressure 76 cm. of mercury) referred to a centimetre and a second as units, we may interpret the result as follows:--Supposing in a mixture of carbon dioxide and air, the density of the carbon dioxide decreases by, say, 1, 2 or 3% of itself in a distance of 1 cm., then the corresponding velocities of the diffusing carbon dioxide will be respectively 0.01, 0.02 and 0.03 times 0.1423, that is, 0.001423, 0.002846 and 0.004269 cm. per second in the three cases.

_Example 2._ If we wished to take a foot and a second as our units, we should have to divide the value of the coefficient of diffusion in Example 1 by the square of the number of centimetres in 1 ft., that is, roughly speaking, by 900, giving the new value of K = 0.00016 roughly.

7. _Numerical Values of the Coefficient of Diffusion._--The table on p. 258 gives the values of the coefficient of diffusion of several of the principal pairs of gases at a pressure of 76 cm. of mercury, and also of a number of other substances. In the gases the centimetre and second are taken as fundamental units, in other cases the centimetre and day.

8. _Irreversible Changes accompanying Diffusion._--The diffusion of two gases at constant pressure and temperature is a good example of an "irreversible process." The gases always tend to mix, never to separate. In order to separate the gases a change must be effected in the external conditions to which the mixture is subjected, either by liquefying one of the gases, or by separating them by diffusion through a membrane, or by bringing other outside influences to bear on them. In the case of liquids, electrolysis affords a means of separating the constituents of a mixture. Every such method involves some change taking place outside the mixture, and this change may be regarded as a "compensating transformation." We thus have an instance of the property that every irreversible change leaves an indelible imprint somewhere or other on the progress of events in the universe. That the process of diffusion obeys the laws of irreversible thermodynamics (if these laws are properly stated) is proved by the fact that the compensating transformations required to separate mixed gases do not essentially involve anything but transformation of energy. The process of allowing gases to mix by diffusion, and then separating them by a compensating transformation, thus constitutes an irreversible cycle, the outside effects of which are that energy somewhere or other must be less capable of transformation than it was before the change. We express this fact by stating that an irreversible process essentially implies a loss of availability. To measure this loss we make use of the laws of thermodynamics, and in particular of Lord Kelvin's statement that "It is impossible by means of inanimate material agency to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects."

+-------------------------------------------+---------+---------------------+--------------+ | Substances. | Temp. | K. | Author. | +-------------------------------------------+---------+---------------------+--------------+ | Carbon dioxide and air | 0°C. | 0.1423 cm²/sec. | J. Loschmidt.| | " " hydrogen | 0°C. | 0.5558 " | " | | " " oxygen | 0°C. | 0.1409 " | " | | " " carbon monoxide | 0°C. | 0.1406 " | " | | " " marsh gas (methane) | 0°C. | 0.1586 " | " | | " " nitrous oxide | 0°C. | 0.0983 " | " | | Hydrogen and oxygen | 0°C. | 0.7214 " | " | | " " carbon monoxide | 0°C. | 0.6422 " | " | | " " sulphur dioxide | 0°C. | 0.4800 " | " | | Oxygen and carbon monoxide | 0°C. | 0.1802 " | " | | Water and ammonia | 20°C. | 1.250 " | G. Hüfner. | | " " | 5°C. | 0.822 " | " | | " common salt (density 1.0269) | | 0.355 cm²/hour. | J. Graham. | | " " " " |14.33°C. | 1.020, 0.996, 0.972,| " | | | | 0.932 cm²/day. | F. Heimbrodt.| | " zinc sulphate (0.312 gm/cm³) | | 0.1162 " | W. Seitz. | | " zinc sulphate (normal) | | 0.2355 " | " | | " zinc acetate (double normal) | | 0.1195 " | " | | " zinc formate (half normal) | | 0.4654 " | " | | " cadmium sulphate (double normal)| | 0.2456 " | " | | " glycerin (1/8n, ½n, 7/8n, 7/8n) |10.14°C. | 0.356, 0.350, 0.342,| F. Heimbrodt.| | | | 0.315 cm²/day. | " | | " urea " " |14.83°C. | 0.973, 0.946, 0.926,| " | | | | 0.883 cm²/day. | " | | " hydrochloric acid |14.30°C. | 2.208, 2.331, | " | | | | 2.480 cm²/day | " | | Gelatin 20% and ammonia | 17°C. | 127.1 " | A. Hagenbach.| | " " carbon dioxide | | 0.845 " | " | | " " nitrous oxide | | 0.509 " | " | | " " oxygen | | 0.230 " | " | | " " hydrogen | | 0.0565 " | " | +-------------------------------------------+---------+---------------------+--------------+

Let us now assume that we have any syste m such as the gases above considered, and that it is in the presence of an indefinitely extended medium which we shall call the "auxiliary medium." If heat be taken from any part of the system, only part of this heat can be converted into work by means of thermodynamic engines; and the rest will be given to the auxiliary medium, and will constitute unavailable energy or waste. To understand what this means, we may consider the case of a condensing steam engine. Only part of the energy liberated by the combustion of the coal is available for driving the engine, the rest takes the form of heat imparted to the condenser. The colder the condenser the more efficient is the engine, and the smaller is the quantity of waste.

The amount of unavailable energy associated with any given transformation is proportional to the absolute temperature of the auxiliary medium. When divided by that temperature the quotient is called the change of "entropy" associated with the given change (see THERMODYNAMICS). Thus if a body at temperature T receives a quantity of heat Q, and if T0 is the temperature of the auxiliary medium, the quantity of work which could be obtained from Q by means of ideal thermodynamic engines would be Q(1 - T0/T), and the balance, which is QT0/T, would take the form of unavailable or waste energy given to the medium. The quotient of this, when divided by T0, is Q/T, and this represents the quantity of entropy associated with Q units of heat at temperature T.

Any irreversible change for which a compensating transformation of energy exists represents, therefore, an increase of unavailable energy, which is measurable in terms of entropy. The increase of entropy is independent of the temperature of the auxiliary medium. It thus affords a measure of the extent to which energy has run to waste during the change. Moreover, when a body is heated, the increase of entropy is the factor which determines how much of the energy imparted to the body is unavailable for conversion into work under given conditions. In all cases we have

increase of unavailable energy ------------------------------- = increase of entropy. temperature of auxiliary medium

When diffusion takes place between two gases inside a closed vessel at uniform pressure and temperature no energy in the form of heat or work is received from without, and hence the entropy gained by the gases from without is zero. But the irreversible processes inside the vessel may involve a gain of entropy, and this can only be estimated by examining by what means mixed gases can be separated, and, in particular, under what conditions the process of mixing and separating the gases could (theoretically) be made reversible.

9. _Evidence derived from Liquefaction of one or both of the Gases._--The gases in a mixture can often be separated by liquefying, or even solidifying, one or both of the components. In connexion with this property we have the important law according to which "The pressure of a vapour in equilibrium with its liquid depends only on the temperature and is independent of the pressures of any other gases or vapours which may be mixed with it." Thus if two closed vessels be taken containing some water and one be exhausted, the other containing air, and if the temperatures be equal, evaporation will go on until the pressure of the vapour in the exhausted vessel is equal to its _partial_ pressure in the other vessel, notwithstanding the fact that the _total_ pressure in the latter vessel is greater by the pressure of the air.

To separate mixed gases by liquefaction, they must be compressed and cooled till one separates in the form of a liquid. If no changes are to take place outside the system, the separate components must be allowed to expand until the work of expansion is equal to the work of compression, and the heat given out in compression is reabsorbed in expansion. The process may be made as nearly reversible as we like by performing the operations so slowly that the substances are practically in a state of equilibrium at every stage. This is a consequence of an important axiom in thermodynamics according to which "any small change in the neighbourhood of a state of equilibrium is to a first approximation reversible."

Suppose now that at any stage of the compression the partial pressures of the two gases are p1 and p2, and that the volume is changed from V to V - dV. The work of compression is (p1 + p2)dV, and this work will be restored at the corresponding stage if each of the separated gases increases in volume from V - dV to V. The ultimate state of the separated gases will thus be one in which each gas occupies the volume V originally occupied by the mixture.

We may now obtain an estimate of the amount of energy rendered unavailable by diffusion. We suppose two gases occupying volumes V1 and V2 at equal pressure p to mix by diffusion, so that the final volume is V1 + V2. Then if before mixing each gas had been allowed to expand till its volume was V1 + V2, work would have been done in the expansion, and the gases could still have been mixed by a reversal of the process above described. In the actual diffusion this work of expansion is lost, and represents energy rendered unavailable at the temperature at which diffusion takes place. When divided by that temperature the quotient gives the increase of entropy. Thus the irreversible processes, and, in particular, the entropy changes associated with diffusion of two gases at uniform pressure, are the same as would take place if each of the gases in turn were to expand by rushing into a vacuum, till it occupied the whole volume of the mixture. A more rigorous proof involves considerations of the thermodynamic potentials, following the methods of J. Willard Gibbs (see ENERGETICS).

Another way in which two or more mixed gases can be separated is by placing them in the presence of a liquid which can freely absorb one of the gases, but in which the other gas or gases are insoluble. Here again it is found by experience that when equilibrium exists at a given temperature between the dissolved and undissolved portions of the first gas, the partial pressure of that gas in the mixture depends on the temperature alone, and is independent of the partial pressures of the insoluble gases with which it is mixed, so that the conclusions are the same as before.

10. _Diffusion through a Membrane or Partition. Theory of the semi-permeable Membrane._--It has been pointed out that diffusion of gases frequently takes place in the interior of solids; moreover, different gases behave differently with respect to the same solid at the same temperature. A membrane or partition formed of such a solid can therefore be used to effect a more or less complete separation of gases from a mixture. This method is employed commercially for extracting oxygen from the atmosphere, in particular for use in projection lanterns where a high degree of purity is not required. A similar method is often applied to liquids and solutions and is known as "dialysis."

In such cases as can be tested experimentally it has been found that a gas always tends to pass through a membrane from the side where its density, and therefore its partial pressure, is greater to the side where it is less; so that for equilibrium the partial pressures on the two sides must be equal. This result is unaffected by the presence of other gases on one or both sides of the membrane. For example, if different gases at the same pressure are separated by a partition through which one gas can pass more rapidly than the other, the diffusion will give rise to a difference of pressure on the two sides, which is capable of doing mechanical work in moving the partition. In evidence of this conclusion Max Planck quotes a test experiment made by him in the Physical Institute of the university of Munich in 1883, depending on the fact that platinum foil at white heat is permeable to hydrogen but impermeable to air, so that if a platinum tube filled with hydrogen be heated the hydrogen will diffuse out, leaving a vacuum.

The details of the experiment may be quoted here:--"A glass tube of about 5 mm. internal diameter, blown out to a bulb at the middle, was provided with a stop-cock at one end. To the other a platinum tube 10 cm. long was fastened, and closed at the end. The whole tube was exhausted by a mercury pump, filled with hydrogen at ordinary atmospheric pressure, and then closed. The closed end of the platinum portion was then heated in a horizontal position by a Bunsen burner. The connexion between the glass and platinum tubes, having been made by means of sealing-wax, had to be kept cool by a continuous current of water to prevent the softening of the wax. After four hours the tube was taken from the flame, cooled to the temperature of the room, and the stop-cock opened under mercury. The mercury rose rapidly, almost completely filling the tube, proving that the tube had been very nearly exhausted."

In order that diffusion through a membrane may be reversible so far as a particular gas is concerned, the process must take place so slowly that equilibrium is set up at every stage (see § 9 above). In order to separate one gas from another consistently with this condition it is necessary that no diffusion of the latter gas should accompany the process. The name "semi-permeable" is applied to an ideal membrane or partition through which one gas can pass, and which offers an insuperable barrier to any diffusion whatever of a second gas. By means of two semi-permeable partitions acting oppositely with respect to two different gases A and B these gases could be mixed or separated by reversible methods. The annexed figure shows a diagrammatic representation of the process.

We suppose the gases contained in a cylindrical tube; P, Q, R, S are four pistons, of which P and R are joined to one connecting rod, Q and S to another. P, S are impermeable to both gases; Q is semi-permeable, allowing the gas A to pass through but not B, similarly R allows the gas B to pass through but not A. The distance PR is equal to the distance QS, so that if the rods are pushed towards each other as far as they will go, P and Q will be in contact, as also R and S. Imagine the space RQ filled with a mixture of the two gases under these conditions. Then by slowly drawing the connecting rods apart until R, Q touch, the gas A will pass into the space PQ, and B will pass into the space RS, and the gases will finally be completely separated; similarly, by pushing the connecting rods together, the two gases will be remixed in the space RQ. By performing the operations slowly enough we may make the processes as nearly reversible as we please, so that no available energy is lost in either change. The gas A being at every instant in equilibrium on the two sides of the piston Q, its density, and therefore its partial pressure, is the same on both sides, and the same is true regarding the gas B on the two sides of R. Also _no work is done in moving the pistons_, for the partial pressures of B on the two sides of R balance each other, consequently, the resultant thrust on R is due to the gas A alone, and is equal and opposite to its resultant thrust on P, so that the connecting rods are at every instant in a state of mechanical equilibrium so far as the pressures of the gases A and B are concerned. We conclude that in the reversible separation of the gases by this method at constant temperature without the production or absorption of mechanical work, the densities and the partial pressures of the two separated gases are the same as they were in the mixture. These conclusions are in entire agreement with those of the preceding section. If this agreement did not exist it would be possible, theoretically, to obtain perpetual motion from the gases in a way that would be inconsistent with the second law of thermodynamics.

Most physicists admit, as Planck does, that it is impossible to obtain an ideal semi-permeable substance; indeed such a substance would necessarily have to possess an infinitely great resistance to diffusion for such gases as could not penetrate it. But in an experiment performed under actual conditions the losses of available energy arising from this cause would be attributable to the imperfect efficiency of the partitions and not to the gases themselves; moreover, these losses are, in every case, found to be completely in accordance with the laws of irreversible thermodynamics. The reasoning in this article being somewhat condensed the reader must necessarily be referred to treatises on thermodynamics for further information on points of detail connected with the argument. Even when he consults these treatises he may find some points omitted which have been examined in full detail at some time or other, but are not sufficiently often raised to require mention in print.

II. _Kinetic Models of Diffusion._--Imagine in the first instance that a very large number of red balls are distributed over one half of a billiard table, and an equal number of white balls over the other half. If the balls are set in motion with different velocities in various directions, diffusion will take place, the red balls finding their way among the white ones, and vice versa; and the process will be retarded by collisions between the balls. The simplest model of a perfect gas studied in the kinetic theory of gases (see MOLECULE) differs from the above illustration in that the bodies representing the molecules move in space instead of in a plane, and, unlike billiard balls, their motion is unresisted, and they are perfectly elastic, so that no kinetic energy is lost either during their free motions, or at a collision.

The mathematical analysis connected with the application of the kinetic theory to diffusion is very long and cumbersome. We shall therefore confine our attention to regarding a medium formed of elastic spheres as a mechanical model, by which the most important features of diffusion can be illustrated. We shall assume the results of the kinetic theory, according to which:--(1) In a dynamical model of a perfect gas the mean kinetic energy of translation of the molecules represents the absolute temperature of the gas. (2) The pressure at any point is proportional to the product of the number of molecules in unit volume about that point into the mean square of the velocity. (The mean square of the velocity is different from but proportional to the square of the mean velocity, and in the subsequent arguments either of these two quantities can generally be taken.) (3) In a gas mixture represented by a mixture of molecules of unequal masses, the mean kinetic energies of the different kinds are equal.

Consider now the problem of diffusion in a region containing two kinds of molecules A and B of unequal mass. The molecules of A in the neighbourhood of any point will, by their motion, spread out in every direction until they come into collision with other molecules of either kind, and this spreading out from every point of the medium will give rise to diffusion. If we imagine the velocities of the A molecules to be equally distributed in all directions, as they would be in a homogeneous mixture, it is obvious that the process of diffusion will be greater, _ceteris paribus_, the greater the velocity of the molecules, and the greater the length of the free path before a collision takes place. If we assume consistently with this, that the coefficient of diffusion of the gas A is proportional to the mean value of Wala, where wa is the velocity and la is the length of the path of a molecule of A, this expression for the coefficient of diffusion is of the right dimensions in length and time. If, moreover, we observe that when diffusion takes place in a fixed direction, say that of the axis of x, it depends only on the resolved part of the velocity and length of path in that direction: this hypothesis readily leads to our taking the mean value of 1/3w_a l_a as the coefficient of diffusion for the gas A. This value was obtained by O. E. Meyer and others.

Unfortunately, however, it makes the coefficients of diffusion unequal for the two gases, a result inconsistent with that obtained above from considerations of the coefficient of resistance, and leading to the consequence that differences of pressure would be set up in different parts of the gas. To equalize these differences of pressure, Meyer assumed that a counter current is set up, this current being, of course, very slow in practice; and J. Stefan assumed that the diffusion of one gas was not affected by collisions between molecules of the _same gas_. When the molecules are mixed in equal proportions both hypotheses lead to the value 1/6([w_a l_a] + [w_b l_b]), (square brackets denoting mean values). When one gas preponderates largely over the other, the phenomena of diffusion are too difficult of observation to allow of accurate experimental tests being made. Moreover, in this case no difference exists unless the molecules are different in size or mass.

Instead of supposing a velocity of translation added after the mathematical calculations have been performed, a better plan is to assume from the outset that the molecules of the two gases have small velocities of translation in opposite directions, superposed on the distribution of velocity, which would occur in a medium representing a gas at rest. When a collision occurs between molecules of different gases a transference of momentum takes place between them, and the quantity of momentum so transferred in one second in a unit of volume gives a dynamical measure of the resistance to diffusion. It is to be observed that, however small the relative velocity of the gases A and B, it plays an all-important part in determining the coefficient of resistance; for without such relative motion, and with the velocities evenly distributed in all directions, no transference of momentum could take place. The coefficient of resistance being found, the motion of each of the two gases may be discussed separately.

One of the most important consequences of the kinetic theory is that if the volume be kept constant the coefficient of diffusion varies as the square root of the absolute temperature. To prove this, we merely have to imagine the velocity of each molecule to be suddenly increased n fold; the subsequent processes, including diffusion, will then go on n times as fast; and the temperature T, being proportional to the kinetic energy, and therefore to the square of the velocity, will be increased n² fold. Thus K, the coefficient of diffusion, varies as [sqrt]T.

The relation of K to the density when the temperature remains constant is more difficult to discuss, but it may be sufficient to notice that if the number of molecules is increased n fold, the chances of a collision are n times as great, and the distance traversed between collisions is (not _therefore_ but as the result of more detailed reasoning) on the average 1/n of what it was before. Thus the free path, and therefore the coefficient of diffusion, varies inversely as the density, or directly as the volume. If the pressure p and temperature T be taken as variables, K varies inversely as p and directly as [sqrt]T³.

Now according to the experiments first made by J. C. Maxwell and J. Loschmidt, it appeared that with constant density K was proportional to T more nearly than to [sqrt]T. The inference is that in this respect a medium formed of colliding spheres fails to give a correct mechanical model of gases. It has been found by L. Boltzmann, Maxwell and others that a system of particles whose mutual actions vary according to the inverse fifth power of the distance between them represents more correctly the relation between the coefficient of diffusion and temperature in actual gases. Other recent theories of diffusion have been advanced by M. Thiesen, P. Langevin and W. Sutherland. On the other hand, J. Thovert finds experimental evidence that the coefficient of diffusion is proportional to molecular velocity in the cases examined of non-electrolytes dissolved in water at 18° at 2.5 grams per litre.

BIBLIOGRAPHY.--The best introduction to the study of theories of diffusion is afforded by O. E. Meyer's Kinetic _Theory of Gases_, translated by Robert E. Baynes (London, 1899). The mathematical portion, though sufficient for ordinary purposes, is mostly of the simplest possible character. Another useful treatise is R. Ruhlmann's _Handbuch der mechanischen Wärmetheorie_ (Brunswick, 1885). For a shorter sketch the reader may refer to J. C. Maxwell's _Theory of Heat_, chaps, xix. and xxii., or numerous other treatises on physics. The theory of the semi-permeable membrane is discussed by M. Planck in his _Treatise on Thermodynamics_, English translation by A. Ogg (1903), also in treatises on thermodynamics by W. Voigt and other writers. For a more detailed study of diffusion in general the following papers may be consulted:--L. Boltzmann, "Zur Integration der Diffusionsgleichung," _Sitzung. der k. bayer. Akad math.-phys. Klasse_ (May 1894); T. des Coudres, "Diffusionsvorgänge in einem Zylinder," _Wied. Ann._ lv. (1895), p. 213; J. Loschmidt, "Experimentaluntersuchungen über Diffusion," _Wien. Sitz._ lxi., lxii. (1870); J. Stefan, "Gleichgewicht und ... Diffusion von Gasmengen," _Wien. Sitz._ lxiii., "Dynamische Theorie der Diffusion," _Wien. Sitz._ lxv. (April 1872); M. Toepler, "Gas-diffusion," _Wied. Ann._ lviii. (1896), p. 599; A. Wretschko, "Experimentaluntersuchungen über die Diffusion von Gasmengen," _Wien. Sitz._ lxii. The mathematical theory of diffusion, according to the kinetic theory of gases, has been treated by a number of different methods, and for the study of these the reader may consult L. Boltzmann, _Vorlesungen über Gastheorie_ (Leipzig, 1896-1898); S. H. Burbury, _Kinetic Theory of Gases_ (Cambridge, 1899), and papers by L. Boltzmann in _Wien. Sitz._ lxxxvi. (1882), lxxxvii. (1883); P. G. Tait, "Foundations of the Kinetic Theory of Gases," _Trans. R.S.E._ xxxiii., xxxv., xxvi., or _Scientific Papers_, ii. (Cambridge, 1900). For recent work reference should be made to the current issues of _Science Abstracts_ (London), and entries under the heading "Diffusion" will be found in the general index at the end of each volume. (G. H. BR.)

DIGBY, SIR EVERARD (1578-1606), English conspirator, son of Everard Digby of Stoke Dry, Rutland, was born on the 16th of May 1578. He inherited a large estate at his father's death in 1592, and acquired a considerable increase by his marriage in 1596 to Mary, daughter and heir of William Mulsho of Gothurst (now Gayhurst), in Buckinghamshire. He obtained a place in Queen Elizabeth's household and as a ward of the crown was brought up a Protestant; but about 1599 he came under the influence of the Jesuit, John Gerard, and soon afterwards joined the Roman Catholics. He supported James's accession and was knighted by the latter on the 23rd of April 1603. In a letter to Salisbury, the date of which has been ascribed to May 1605, Digby offered to go on a mission to the pope to obtain from the latter a promise to prevent Romanist attempts against the government in return for concessions to the Roman Catholics; adding that if severe measures were again taken against them "within brief there will be massacres, rebellions and desperate attempts against the king and state." Digby had suffered no personal injury or persecution on account of his religion, but he sympathized with his co-religionists; and when at Michaelmas, 1605, the government had fully decided to return to the policy of repression, the authors of the Gunpowder Plot (q.v.) sought his financial support, and he joined eagerly in the conspiracy. His particular share in the plan was the organization of a rising in the Midlands; and on the pretence of a hunting party he assembled a body of gentlemen together at Danchurch in Warwickshire on the 5th of November, who were to take action immediately the news arrived from London of the successful destruction of the king and the House of Lords, and to seize the person of the princess Elizabeth, who was residing in the neighbourhood. The conspirators arrived late on the evening of the 6th to tell their story of failure and disaster, and Digby, who possibly might have escaped the more serious charge of high treason, was persuaded by Catesby, with a false tale that the king and Salisbury were dead, to further implicate himself in the plot and join the small band of conspirators in their hopeless endeavour to raise the country. He accompanied them, the same day, to Huddington in Worcestershire and on the 7th to Holbeche in Staffordshire. The following morning, however, he abandoned his companions, dismissed his servants except two, who declared "they would never leave him but against their will," and attempted with these to conceal himself in a pit. He was, however, soon discovered and surrounded. He made a last effort to break through his captors on horseback, but was taken and conveyed a prisoner to the Tower. His trial took place in Westminster Hall, on the 27th of January 1606, and alone among the conspirators he pleaded guilty, declaring that the motives of his crime had been his friendship for Catesby and his devotion to his religion. He was condemned to death, and his execution, which took place on the 31st, in St Paul's Churchyard, was accompanied by all the brutalities exacted by the law.

Digby was a handsome man, of fine presence. Father Gerard extols his skill in sport, his "riding of great horses," as well as his skill in music, his gifts of mind and his religious devotion, and concludes "he was as complete a man in all things, that deserved estimation or might win affection as one should see in a kingdom." Some of Digby's letters and papers, which include a poem before his execution, a last letter to his infant sons and correspondence with his wife from the Tower, were published in _The Gunpowder Treason_ by Thomas Barlow, bishop of Lincoln, in 1679. He left two sons, of whom the elder, Sir Kenelm Digby, was the well-known author and diplomatist.

See works on the Gunpowder Plot; Narrative of Father Gerard, in _Condition of the Catholics under James I._ by J. Morris (1872), &c. A life of Digby under the title of _A Life of a Conspirator_, by a Romish Recusant (Thomas Longueville), was published in 1895. (P. C. Y.)

DIGBY, SIR KENELM (1603-1665), English author, diplomatist and naval commander, son of Sir Everard Digby (q.v.), was born on the 11th of July 1603, and after his father's execution in 1606 resided with his mother at Gayhurst, being brought up apparently as a Roman Catholic. In 1617 he accompanied his cousin, Sir John Digby, afterwards 1st earl of Bristol, and then ambassador in Spain, to Madrid. On his return in April 1618 he entered Gloucester Hall (now Worcester College), Oxford, and studied under Thomas Allen (1542-1632), the celebrated mathematician, who was much impressed with his abilities and called him the _Mirandula_, i.e. the infant prodigy, of his age.[1] He left the university without taking a degree in 1620, and travelled in France, where, according to his own account, he inspired an uncontrollable passion in the queen-mother, Marie de' Medici, now a lady of more than mature age and charms; he visited Florence, and in March 1623 joined Sir John Digby again at Madrid, at the time when Prince Charles and Buckingham arrived on their adventurous expedition. He joined the prince's household and returned with him to England on the 5th of October 1623, being knighted by James I. on the 23rd of October and receiving the appointment of gentleman of the privy chamber to Prince Charles. In 1625 he married secretly Venetia, daughter of Sir Edward Hanley of Tonge Castle, Shropshire, a lady of extraordinary beauty and intellectual attainments, but of doubtful virtue. Digby was a man of great stature and bodily strength. Edward Hyde, afterwards earl of Clarendon, who with Ben Jonson was included among his most intimate friends, describes him as "a man of very extraordinary person and presence which drew the eyes of all men upon him, a wonderful graceful behaviour, a flowing courtesy and civility, and such a volubility of language as surprised and delighted."[2] Digby for some time was excluded from public employment by Buckingham's jealousy of his cousin, Lord Bristol. At length in 1627, on the latter's advice, Digby determined to attempt "some generous action," and on the 22nd of December, with the approval of the king, embarked as a privateer with two ships, with the object of attacking the French ships in the Venetian harbour of Scanderoon. On the 18th of January he arrived off Gibraltar and captured several Spanish and Flemish vessels. From the 15th of February to the 27th of March he remained at anchor off Algiers on account of the sickness of his men, and extracted a promise from the authorities of better treatment of the English ships. He seized a rich Dutch vessel near Majorca, and after other adventures gained a complete victory over the French and Venetian ships in the harbour of Scanderoon on the 11th of June. His successes, however, brought upon the English merchants the risk of reprisals, and he was urged to depart. He returned home in triumph in February 1629, and was well received by the king, and was made a commissioner of the navy in October 1630, but his proceedings were disavowed on account of the complaints of the Venetian ambassador. In 1633 Lady Digby died, and her memory was celebrated by Ben Jonson in a series of poems entitled _Eupheme_, and by other poets of the day. Digby retired to Gresham College, and exhibited extravagant grief, maintaining a seclusion for two years. About this time Digby professed himself a Protestant, but by October 1635, while in France, he had already returned to the Roman Catholic faith.[3] In a letter dated the 27th of March 1636 Laud remonstrates with him, but assures him of the continuance of his friendship.[4] In 1638 he published _A Conference with a Lady about choice of a Religion_, in which he argues that the Roman Church, possessing alone the qualifications of universality, unity of doctrine and uninterrupted apostolic succession, is the only true church, and that the intrusion of error into it is impossible. The same subject is treated in letters to George Digby, afterwards 2nd earl of Bristol, dated the 2nd of November 1638 and the 29th of November 1639, which were published in 1651, as well as in a further _Discourse concerning Infallibility in Religion_ in 1652. Returning to England he associated himself with the queen and her Roman Catholic friends, and joined in the appeal to the English Romanists for money to support the king's Scottish expedition.[5] In consequence he was summoned to the bar of the House of Commons on the 27th of January 1641, and the king was petitioned to remove him with other recusants from his councils. He left England, and while at Paris killed in a duel a French lord who had insulted Charles I. in his presence. Louis XIII. took his part, and furnished him with a military escort into Flanders. Returning home he was imprisoned, by order of the House of Commons, early in 1642, successively in the "Three Tobacco Pipes nigh Charing Cross," where his delightful conversation is said to have transformed the prison into "a place of delight,"[6] and at Winchester House. He was finally released and allowed to go to France on the 30th of July 1643, through the intervention of the queen of France, Anne of Austria, on condition that he would neither promote nor conceal any plots abroad against the English government.

Before leaving England an attempt was made to draw from him an admission that Laud, with whom he had been intimate, had desired to be made a cardinal, but Digby denied that the archbishop had any leanings towards Rome. On the 1st of November 1643 it was resolved by the Commons to confiscate his property. He published in London the same year _Observations on the 22nd stanza in the 9th canto of the 2nd book of Spenser's "Faërie Queene,"_ the MS. of which is in the Egerton collection (British Museum, No. 2725 f. 117 b), and _Observations_ on a surreptitious and unauthorized edition of the _Religio Medici_, by Sir Thomas Browne, from the Roman Catholic point of view, which drew a severe rebuke from the author. After his arrival in Paris he published his chief philosophical works, _Of Bodies_ and _Of the Immortality of Man's Soul_ (1644), autograph MSS. of which are in the Bibliothèque Ste Geneviève at Paris, and made the acquaintance of Descartes. He was appointed by Queen Henrietta Maria her chancellor, and in the summer of 1645 he was despatched by her to Rome to obtain assistance. Digby promised the conversion of Charles and of his chief supporters. At first his eloquence made a great impression. Pope Innocent X. declared that he spoke not merely as a Catholic but as an ecclesiastic. But the absence of any warrant from Charles himself roused suspicions as to the solidity of his assurances, and he obtained nothing but a grant of 20,000 crowns. A violent quarrel with the pope followed, and he returned in 1646, having consented in the queen's name to complete religious freedom for the Roman Catholics, both in England and Ireland, to an independent parliament in Ireland, and to the surrender of Dublin and all the Irish fortresses into the hands of the Roman Catholics, the king's troops to be employed in enforcing the articles and the pope granting about £36,000 with a promise of further payments in obtaining direct assistance. In February 1649 Digby was invited to come to England to arrange a proposed toleration of the Roman Catholics, but on his arrival in May the scheme had already been abandoned. He was again banished on the 31st of August, and it was not till 1654 that he was allowed by the council of state to return. He now entered into close relations with Cromwell, from whom he hoped to obtain toleration for the Roman Catholics, and whose alliance he desired to secure for France rather than for Spain, and was engaged by Cromwell, much to the scandal of both Royalists and Roundheads, in negotiations abroad, of which the aim was probably to prevent a union between those two foreign powers. He visited Germany, in 1660 was in Paris, and at the Restoration returned to England. He was well received in spite of his former relations with Cromwell, and was confirmed in his post as Queen Henrietta Maria's chancellor. In January 1661 he delivered a lecture, which was published the same month, at Gresham College, on the vegetation of plants, and became an original member of the Royal Society in 1663. In January 1664 he was forbidden to appear at court, the cause assigned being that he had interposed too far in favour of the 2nd earl of Bristol, disgraced by the king on account of the charge of high treason brought by him against Clarendon into the House of Lords. The rest of his life was spent in the enjoyment of literary and scientific society at his house in Covent Garden. He died on the 11th of June 1665. He had five children, of whom two, a son and one daughter, survived him.

Digby, though he possessed for the time a considerable knowledge of natural science, and is said to have been the first to explain the necessity of oxygen to the existence of plants, bears no high place in the history of science. He was a firm believer in astrology and alchemy, and the extraordinary fables which he circulated on the subject of his discoveries are evidence of anything rather than of the scientific spirit. In 1656 he made public a marvellous account of a city in Tripoli, petrified in a few hours, which he printed in the _Mercurius Politicus_. Malicious reports had been current that his wife had been poisoned by one of his prescriptions, viper wine, taken to preserve her beauty. Evelyn, who visited him in Paris in 1651, describes him as an "errant mountebank." Henry Stubbes characterizes him as "the very Pliny of our age for lying," and Lady Fanshawe refers to the same "infirmity."[7] His famous "powder of sympathy," which seems to have been only powder of "vitriol," healed without any contact, by being merely applied to a rag or bandage taken from the wound, and Digby records a miraculous cure by this means in a lecture given by him at Montpellier on this subject in 1658, published in French and English the same year, in German in 1660 and in Dutch in 1663; but Digby's claim to its original discovery is doubtful, Nathaniel Highmore in his _History of Generation_ (1651, p. 113) calling the powder "Talbot's powder," and ascribing its invention to Sir Gilbert Talbot. Some of Digby's pills and preparations, however, described in _The Closet of the Eminently Learned Sir Kenelm Digby Knt. Opened_ (publ. 1677), are said to make less demand upon the faith of patients, and his injunction on the subject of the making of tea, to let the water "remain upon it no longer than you can say the Miserere Psalm very leisurely," is one by no means to be ridiculed. As a philosopher and an Aristotelian Digby shows little originality and followed the methods of the schoolmen. His Roman Catholic orthodoxy mixed with rationalism, and his political opinions, according to which any existing authority should receive support, were evidently derived from Thomas White (1582-1676), the Roman Catholic philosopher, who lived with him in France. White published in 1651 _Institutionum Peripateticorum libri quinque_, purporting to expound Digby's "peripatetic philosophy," but going far beyond Digby's published treatises. Digby's _Memoirs_ are composed in the high-flown fantastic manner then usual when recounting incidents of love and adventure, but the style of his more sober works is excellent. In 1632 he presented to the Bodleian library a collection of 236 MSS., bequeathed to him by his former tutor Thomas Allen, and described in _Catalogi codicum manuscriptorum bibliothecae Bodleianae_, by W. D. Macray, part ix. Besides the works already mentioned Digby translated _A Treatise of adhering to God written by Albert the Great, Bishop of Ratisbon_ (1653); and he was the author of _Private Memoirs_, published by Sir N. H. Nicholas from _Harleian MS. 6758_ with introduction (1827); _Journal of the Scanderoon Voyage in 1628_, printed by J. Bruce with preface (Camden Society, 1868); _Poems from Sir Kenelm Digby's Papers_... with preface and notes (Roxburghe Club, 1877); in the _Add. MSS._ 34,362 f. 66 is a poem _Of the Miserys of Man_, probably by Digby; _Choice of Experimental Receipts in Physick and Chirurgery_ ... _collected by Sir K. Digby_ (1668), and _Chymical Secrets and Rare Experiments_ (1683), were published by G. Hartman, who describes himself as Digby's steward and laboratory assistant.

See the _Life of Sir Kenelm Digby by one of his Descendants_ (T. Longueville), 1896. (P. C. Y.)

FOOTNOTES:

[1] _Letters by Eminent Persons_ (Aubrey's Lives), ii. 324.

[2] _Life and Continuation._

[3] Strafford's _Letters_, i. 474.

[4] Laud's _Works_, vi. 447.

[5] _Thomason Tracts_, Brit. Mus. E 164 (15).

[6] _Archaeologia Cantiana_, ii. 190.

[7] _Dict. of Nat. Biog._ sub "Digby." See also Robert Boyle's _Works_ (1744), v. 302.

DIGBY, KENELM HENRY (1800-1880), English writer, youngest son of William Digby, dean of Clonfert, was born at Clonfert, Ireland, in 1800. He was educated at Trinity College, Cambridge, and soon after taking his B.A. degree there in 1819 became a Roman Catholic. He spent most of his life, which was mainly devoted to literary pursuits, in London, where he died on the 22nd of March 1880. Digby's reputation rests chiefly on his earliest publication, _The Broadstone of Honour, or Rules for the Gentlemen of England_ (1822), which contains an exhaustive survey of medieval customs, full of quotations from varied sources. The work was subsequently enlarged and issued (1826-1827) in four volumes entitled: _Godefridus_, _Tancredus_, _Morus_ and _Orlandus_ (numerous re-impressions, the best of which is the edition brought out by B. Quaritch in five volumes, 1876-1877).

Among Digby's other works are: _Mores Catholici, or Ages of Faith_ (11 vols., London, 1831-1840); _Compitum; or the Meeting of the Ways at the Catholic Church_ (7 vols., London, 1848-1854); _The Lovers' Seat, Kathemérina; or Common Things in relation to Beauty, Virtue and Faith_ (2 vols., London, 1856). A complete list is given in J. Gillow's _Bibliographical Dictionary of English Catholics_, ii. 81-83.

DIGENES ACRITAS, BASILIUS, Byzantine national hero, probably lived in the 10th century. He is named Digenes (of double birth) as the son of a Moslem father and a Christian mother; Acritas ([Greek: akra], frontier, boundary), as one of the frontier guards of the empire, corresponding to the Roman _milites limitanei_. The chief duty of these _acritae_ consisted in repelling Moslem inroads and the raids of the _apelatae_ (cattle-lifters), brigands who may be compared with the more modern Klephts. The original Digenes epic is lost, but four poems are extant, in which the different incidents of the legend have been worked up by different hands. The first of these consists of about 4000 lines, written in the so-called "political" metre, and was discovered in the latter part of the 19th century, in a 16th-century MS., at Trebizond; the other three MSS. were found at Grotta Ferrata, Andros and Oxford. The poem, which has been compared with the _Chanson de Roland_ and the _Romance of the Cid_, undoubtedly contains a kernel of fact, although it cannot be regarded as in any sense an historical record. The scene of action is laid in Cappadocia and the district of the Euphrates.

Editions of the Trebizond MS. by C. Sathas and E. Legrand in the _Collection des monuments pour servir à l'étude de la langue néohellénique_, new series, vi. (1875), and by S. Joannides (Constantinople, 1887). See monographs by A. Luber (Salzburg, 1885) and G. Wartenberg (Berlin, 1897). Full information will be found in C. Krumbacher's _Geschichte der byzantinischen Litteratur_, p. 827 (2nd ed., 1897); see also G. Schlumberger, _L'Épopée Byzantine à la fin du dixième siècle_ (1897).

DIGEST, a term used generally of any digested or carefully arranged collection or compendium of written matter, but more particularly in law of a compilation in condensed form of a body of law digested in a systematical method; e.g. the Digest (_Digesta_) or Pandects ([Greek: Pandektai]) of Justinian, a collection of extracts from the earlier jurists compiled by order of the emperor Justinian. The word is also given to the compilations of the main points (marginal or hand-notes) of decided cases, usually arranged in alphabetical and subject order, and published under such titles as "Common Law Digest," "Annual Digest," &c.

DIGESTIVE ORGANS (PATHOLOGY). Several facts of importance have to be borne in mind for a proper appreciation of the pathology of the organs concerned in digestive processes (for the anatomy see ALIMENTARY CANAL and allied articles). In the first place, more than all other systems, the digestive comprises greater range of structure and exhibits wider diversity of function within its domain. Each separate structure and each different function presents special pathological signs and symptoms. Again, the duties imposed upon the system have to be performed notwithstanding constant variations in the work set them. The crude articles of diet offered them vary immensely in nature, bulk and utility, from which they must elaborate simple food-elements for absorption, incorporate them after absorption into complex organic substances properly designed to supply the constant needs of cellular activity, of growth and repair, and fitly harmonized to fulfil the many requirements of very divergent processes and functions. Any form of unphysiological diet, each failure to cater for the wants of any special tissue engaged in, or of any processes of, metabolism, carry with them pathological signs. Perhaps in greater degree than elsewhere are the individual sections of the digestive system dependent upon, and closely correlated with, one another. The lungs can only yield oxygen to the blood when the oxygen is uncombined; no compounds are of use. The digestive organs have to deal with an enormous variety of compound bodies, from which to obtain the elements necessary for protoplasmic upkeep and activity. Morbid lesions of the respiratory and circulatory systems are frequently capable of compensation through increased activity elsewhere, and the symptoms they give rise to follow chiefly along one line; diseases of the digestive organs are more liable to occasion disorders elsewhere than to excite compensatory actions. The digestive system includes every organ, function and process concerned with the utilization of food-stuffs, from the moment of their entrance into the mouth, their preparation in the canal, assimilation with the tissues, their employment therein, up to their excretion or expulsion in the form of waste. Each portion resembles a link of a continuous chain; each link depends upon the integrity of the others, the weakening or breaking of one straining or making impotent the chain as a whole.

The mucous membrane lining the alimentary tract is the part most subject to pathological alterations, and in this connexion it should be remembered that this membrane differs both in structure and functions throughout the tract. Chiefly protective from the mouth to the cardia, it is secretory and absorbent in the stomach and bowel; while the glandular cells forming part of it secrete both acid and alkaline fluids, several ferments or mucus. Over the dorsum of the tongue its modified cells subserve the sense of taste. Without, connected with it by the submucous connective tissue, is placed the muscular coat, and externally over the greater portion of its length the peritoneal serous membrane. All parts are supplied with blood-vessels, lymph-ducts and nerves, the last belonging either to local or to central circuits. Associated with the tract are the salivary glands, the liver and the pancreas; while, in addition, lymphoid tissue is met with diffusely scattered throughout the lining membranes in the tonsils, appendix, solitary glands and Peyer's patches, and the mesenteric glands. The functions of the various parts of the system in whose lesions we are here interested are many in number, and can only be summarized here. (For the physiology of digestion see NUTRITION.) Broadly, they maybe given as: (1) Ingestion and swallowing of food, transmission of it through the tract, and expulsion of the waste material; (2) secretion of acids and alkalis for the performance of digestive processes, aided by (3) elaboration and addition of complex bodies, termed enzymes or ferments; (4) secretion of mucus; (5) protection of the body against organismal infection, and against toxic products; (6) absorption of food elements and reconstitution of them into complex substances fitted for metabolic application; and (7) excretion of the waste products of protoplasmic action. These functions may be altered by disease, singly or in conjunction; it is rare, however, to find but one affected, while an apparently identical disturbance of function may often arise from totally different organic lesions. Another point of importance is seen in the close interdependence which exists between the secretions of acid and those of alkaline reaction. The difference in reaction seems to act _mutatis mutandis_ as a stimulant in each instance.

_General Diseases._

Vascular lesions.

In all sections of the alimentary canal actively engaged in the digestion of food, a well-marked local engorgement of the blood-vessels supplying the walls occurs. The hyperaemia abates soon after completion of the special duties of the individual sections. This normal condition may be abnormally exaggerated by overstimulation from irritant poisons introduced into the canal; from too rich, too copious or indigestible articles of diet; or from too prolonged an experience of some unvaried kind of food-stuff, especially if large quantities of it are necessary for metabolic needs; entering into the first stage of inflammation, acute hyperaemia. More important, because productive of less tractable lesions, is passive congestion of the digestive organs. Whenever the flow of blood into the right side of the heart is hindered, whether it arise from disease of the heart itself, or of the lungs, or proceed from obstruction in some part of the portal system, the damming-back of the venous circulation speedily produces a more or less pronounced stasis of the blood in the walls of the alimentary canal and in the associated abdominal glands. The lack of a sufficiently vigorous flow of blood is followed by deficient secretion of digestive agents from the glandular elements involved, by decreased motility of the muscular coats of the stomach and bowel, and lessened adaptability throughout for dealing with even slight irregular demands on their powers. The mucous membrane of the stomach and bowel, less able to withstand the effects of irritation, even of a minor character, readily passes into a condition of chronic catarrh, while it frequently is the seat of small abrasions, haemorrhagic erosions, which may cause vomiting of blood and the appearance of blood in the stools. Obstruction to the flow of blood from the liver leads to dilatation of its blood-vessels, consequent pressure upon the hepatic cells adjoining them, and their gradual loss of function, or even atrophy and degeneration. In addition to the results of such passive congestion exhibited by the stomach and bowel as noted above, passive congestion of the liver is often accompanied by varicose enlargement of the abdominal veins, in particular of those which surround the lower end of the oesophagus, the lowest part of the rectum and anus. In the latter position these dilated veins constitute what are known as haemorrhoids or piles, internal or external as their site lies within or outside the anal aperture.

Inflammatory lesions.

The mucous and serous membranes of the canal and the glandular elements of the associated organs are the parts most subject to inflammatory affections. Among the several sections of the digestive tract itself, the oesophagus and jejunum are singularly exempt from inflammatory processes; the fauces, stomach, caecum and appendix, ileum, mouth and duodenum (including the opening of the common bile-duct), are more commonly involved. _Stomatitis_, or inflammation of the mouth, has many predisposing factors, but it has now been definitely determined that its exciting cause is always some form of micro-organism. Any condition favouring oral sepsis, as carious teeth, pyorrhoea alveolaris (a discharge of pus due to inflamed granulations round carious teeth), granulations beneath thick crusts of tartar, or an irritating tooth plate, favours the growth of pyogenic organisms and hence of stomatitis. Many varieties of this disease have been described, but all are forms of "pyogenic" or "septic stomatitis." This in its mildest form is catarrhal or erythematous, and is attended only by slight swelling tenderness and salivation. In its next stage of acuteness it is known as "membranous," as a false membrane is produced somewhat resembling that due to diphtheria, though caused by a staphylococcus only. A still more acute form is "ulcerative," which may go on to the formation of an abscess beneath the tongue. Scarlet fever usually gives rise to a slight inflammation of the mouth followed by desquamation, but more rarely it is accompanied by a most severe oedematous stomatitis with glossitis and tonsillitis. Erysipelas on the face may infect the mouth, and an acute stomatitis due to the diphtheria bacillus, Klebs-Loeffler bacillus, has been described. A distinct and very dangerous form of stomatitis in infants and young children is known as "aphthous stomatitis" or "thrush." This is caused by the growth of _Oidium albicans_. It is always preceded by a gastro-enteritis and dry mouth, and if this is not attended to, soon attracts attention by the little white raised patches surrounded by a dusky red zone scattered on tongue and cheeks. Epidemics have occurred in hospitals and orphanages. Mouth breathing is the cause of many ills. As a result of this, the mucous membrane of the tongue, &c., becomes dry, micro-organisms multiply and the mouth becomes foul. Also from disease of the nose, the upper jaw, palate and teeth do not make proper progress in development. There is overgrowth of tonsils, and adenoids, with resulting deafness, and the child's mental development suffers. An ordinary "sore throat" usually signifies acute catarrh of the fauces, and is of purely organismal origin, "catching cold" being only a secondary and minor cause. In "relaxed throats" there is a chronic catarrhal state of the lining membrane, with some passive congestion. The tonsils are peculiarly liable to catarrhal attacks, as might a priori be expected by reason of their Cerberus-like function with regard to bacterial intruders. Still, acute attacks of tonsillitis appear on good evidence to be more common among individuals predisposed constitutionally to rheumatic manifestations. Cases of acute tonsillitis may or may not go on to suppuration or quinsy; in all there is great congestion of the glands, increased mucus secretion, and often secondary involvement of the lymphatic glands of the neck. Repeated acute attacks often lead to chronic inflammation, in which the glands are enlarged, and often hypertrophied in the true sense of the term. The oesophagus is the seat of inflammation but seldom. In infants and young children thrush due to _Oidium albicans_ may spread from the mouth, and also a diphtheritic inflammation spreads from the fauces into the oesophagus. A catarrhal oesophagitis is rarely seen, but the commonest form is traumatic, due to the swallowing of boiling water, corrosive or irritant substances, &c. A non-malignant ulceration may result which later leads on to an oesophageal stricture. The physical changes presented by the coats of the stomach and the intestine, the subjects of catarrhal attacks, closely resemble one another, but differ symptomatically. Acute catarrh of the stomach is associated with intense hyperaemia of its lining coats, with visible engorgement and swelling of the mucous membrane, and an excessive secretion of mucus. The formation of active gastric juice is arrested, digestion ceases, peristaltic movements are sluggish or absent, unless so over-stimulated that they act in a direction the reverse of the normal, and induce expulsion of the gastric contents by vomiting. The gastric contents, in whatever degree of dilution or concentration they may have been ingested, when ejected are of porridge-thick consistency, and often but slightly digested. Such conditions may succeed a severe alcoholic bout, be caused by irritant substances taken in by the mouth or arise from fermentative processes in the stomach contents themselves. Should the irritating material succeed in passing from the stomach into the bowel, similar physical signs are present; but as the quickest path offered for the expulsion of the offending substances from the body is downwards, peristalsis is increased, the flow of fluid from the intestinal glands is larger in bulk, though of less potency as regards its normal actions, than in health, and diarrhoea, with removal of the irritant, follows. As a general rule, the more marked the involvement of the large bowel, the severer and more fluid is the resultant diarrhoea. Inflammation of the stomach may be due to mechanical injury, thermal or chemical irritants or invasion by micro-organisms. Also all the symptoms of gastric catarrh may be brought on by any acute emotion. The commonest mechanical injury is that due to an excess of food, especially when following on a fast; poisons act as irritants, and also the weevils of cheese and the larvae of insects.

Inflammatory affections of the caecum and its attached appendix vermiformis are very common, and give rise to several special symptoms and signs. Acute inflammatory appendicitis appears to be increasing in frequency, and is associated by many with the modern deterioration in the teeth. Constipation certainly predisposes to it, and it appears to be more prevalent among medical men, commercial travellers, or any engaged in arduous callings, subjected to irregular meals, fatigue and exposure. A foreign body is the exciting cause in many cases, though less commonly so than was formerly imagined. The inflammation in the appendix varies in intensity from a very slight catarrhal or simple form to an ulcerative variety, and much more rarely to the acute fulminating appendicitis in which necrosis of the appendix with abscess formation occurs. It is always accompanied by more or less peritonitis, which is protective in nature, shutting in the inflammatory process. Very similar symptomatically is the condition termed perityphlitis, doubtless in former days frequently due to the appendix, an acute or chronic inflammation of the walls of the caecum often leading to abscess formation outside the gut, with or without direct communication with the canal. The colon is subject to three main forms of inflammation. In simple _colitis_ the mucous membrane of the colon is intensely injected, bright red in colour, and secreting a thick mucus, but there is no accompanying ulceration. It is often found in association with some constitutional disease, as Bright's disease, and also with cancer of the bowel. But when it has no association with other trouble it is probably bacterial in origin, the _Bacillus enteritidis spirogenes_ having been isolated in many cases. The motions always contain large quantities of mucus and more or less blood. A second very severe form of inflammation of the colon is known as "membranous colitis," and this may be either dyspeptic, or secondary to other diseases. In this trouble membranes are passed _per anum_, accompanied by a pain so intense as often to cause fainting. In severe cases complete tubular casts of the intestine have been found. Often the motions contain very little faecal matter, but consist only of membranes, mucus and a little blood. A third form is that known as "ulcerative colitis." Any part of the large intestine may be affected, and the ulceration shows no special distribution. In severe cases the muscular coat is exposed, and perforation may ensue. The number of ulcers varies from a few to many dozen, and in size from a pea to a five-shilling piece. Like all chronic intestinal ulcers they show a tendency to become transverse.

Chronic catarrhal affections of the stomach are very common, and often follow upon repeated acute attacks. In them the connective tissue increases at the expense of the glandular elements; the mucous membrane becomes thickened and less active in function. Should the muscular coat be involved, the elasticity and contractility of the organ suffer; peristaltic movement is weakened; expulsion of the contents through the pylorus hindered; and, aggravated by these effects, the condition becomes worse, atonic dyspepsia in its most pronounced form results, with or without dilatation. Chronic vascular congestion may occasion in process of time similar signs and symptoms.

Duodenal catarrh is constantly associated with jaundice, indeed is most probably the commonest cause of catarrhal jaundice; often it is accompanied by catarrh of the common bile-duct. Chronic inflammation of the small intestine gives rise to less prominent symptoms than in the stomach. It generally arises from more than one cause; or rather secondary causes rapidly become as important as the primary in its incidence. Chronic congestion and prolonged irritation lead to deficient secretion and sluggish peristalsis; these effects encourage intestinal putrefaction and auto-intoxication; and these latter, in turn, increase the local unrest.

Infective lesions.

The intestinal mucous membrane, the peritoneum and the mesenteric glands are the chief sites of tubercular infection in the digestive organs. Rarely met with in the gullet and stomach, and comparatively seldom in the mouth and lips, tubercular inflammation of the small intestine and peritoneum is common. Tubercular enteritis is a frequent accompaniment of phthisis, but may occur apart from tubercle of other organs. Children are especially subject to the primary form. Tubercular peritonitis often is present also. The inflammatory process readily tends towards ulcer formation, with haemorrhage and sometimes perforation. If in the large bowel, the symptoms are usually less acute than those characterizing tubercular inflammation of the small intestine. The appendix has been found to be the seat of tubercular processes; in the rectum they form the general cause of the fistulae and abscesses so commonly met with here. Tubercular peritonitis may be primary or secondary, acute or chronic; occasionally very acute cases are seen running a rapid course; the majority are chronic in type. The tubercles spread over the surface of the serous membrane, and if small and not very numerous may give rise in chronic cases to few symptoms; if larger, and especially when they involve and obstruct the lymph- and blood-vessels, ascites follows. It is hardly possible that tubercular invasion of the mesenteric glands can ever occur unaccompanied by peritoneal infection; but when the infection of the glands constitutes the most prominent sign, the term _tabes mesenterica_ is sometimes employed. Here the glands, enlarged, form a doughy mass in the abdomen, leading to marked protrusion of the abdominal walls, with wasting elsewhere and diarrhoea.

The liver is seldom attacked by tubercle, unless in cases of general miliary tuberculosis. Now and then it contains large caseous tubercular masses in its substance.

An important fact with regard to the tubercular processes in the digestive organs lies in the ready response to treatment shown by many cases of peritoneal or mesenteric invasion, particularly in the young.

The later sequelae of syphilis display a predilection for the rectum and the liver, usually leading to the development of a stricture in the former, to a diffuse hepatitis or the formation of gummata in the second. In inherited syphilis the temporary teeth usually appear early, are discoloured and soon crumble away. The permanent teeth may be sound and healthy, but are often--especially the upper incisors--notched and stunted, when they are known as "Hutchinson's teeth." As the result both of syphilis and of tubercle, the tissues of the liver and bowel may present a peculiar alteration; they become amyloid, or lardaceous, a condition in which they appear "waxy," are coloured dark mahogany brown with dilute iodine solutions, and show degenerative changes in the connective tissue.

The _Bacillus typhosus_ discovered by Eberth is the causal agent of typhoid fever, and has its chief seat of activity in the small intestine, more especially in the lower half of the ileum. Attacking the lymphoid follicles in the mucous membrane, it causes first inflammatory enlargement, then necrosis and ulceration. The adjacent portions of the mucous membrane show acute catarrhal changes. Diarrhoea, of a special "pea-soup" type, may or may not be present; while haemorrhage from the bowel, if ulcers have formed, is common. As the ulcers frequently extend down to the peritoneal coat of the bowel, perforation of this membrane and extravasation into the peritoneal cavity is easily induced by irritants introduced into or elaborated in the bowel, acting physically or by the excitation of hyper-peristalsis.

True Asiatic cholera is due to the comma-bacillus or spirillum of cholera, which is found in the rice-water evacuations, in the contents of the intestine after death, and in the mucous membrane of the intestine just beneath the epithelium. It has not been found in the blood. It produces an intense irritation of the bowel, seldom of the stomach, without giving rise locally to any marked physical change; it causes violent diarrhoea and copious discharges of "rice-water" stools, consisting largely of serum swarming with the organism.

Dysentery gives rise to an inflammation of the large intestine and sometimes of the lower part of the ileum, resulting in extensive ulceration and accompanied by faecal discharges of mucus, muco-pus or blood. In some forms a protozoan, the _Amoeba dysenteriae_, is found in the stools--this is the amoebic dysentery; in other cases a bacillus, _Bacillus dysenteriae_, is found--the bacillary dysentery.

Acute parotitis, or mumps, is an infectious disease of the parotid glands, chiefly interesting because of the association between it and the testes in males, inflammation of these glands occasionally following or replacing the affection of the parotids. The causal agent is probably organismal, but has as yet escaped detection.

New growths.

The relative frequency with which malignant growths occur in the different organs of the digestive system may be gathered from the tabular analysis, on p. 266, of 1768 cases recorded in the books of the Edinburgh Royal Infirmary as having been treated in the medical and surgical wards between the years 1892 and 1899 inclusive. Of these, 1263, or 71.44%, were males; 505, or 28.56%, females. (See Table I. p. 266.)

If the figures there given be classified upon broader lines, the results are as given in Table II. p. 266, and speak for themselves.

The digestive organs are peculiarly subject to malignant disease, a result of the incessant changes from passive to active conditions, and vice versa, called for by repeated introduction of food; while the comparative frequency with which different parts are attacked depends, in part, upon the degree of irritation or changes of function imposed upon them. Scirrhous, encephaloid and colloid forms of carcinoma occur. In the stomach and oesophagus the scirrhous form is most common, the soft encephaloid form coming next. The most common situation for cancerous growth in the stomach is the pyloric region. Walsh out of 1300 cases found 60.8% near the pylorus, 11.4% over the lesser curvature, and 4.7% more or less over the whole organ. The small intestine is rarely attacked by cancer; the large intestine frequently. The rectum, sigmoid flexure, caecum and colon are affected, and in this order, the cylindrical-celled form being the most common. Carcinoma of the peritoneum is generally colloid in character, and is often secondary to growths in other organs. Cancer of the liver follows cancer of the stomach and rectum in frequency of occurrence, and is relatively more common in females than males. Secondary invasion of the liver is a frequent sequel to gastric cancer. The pancreas occasionally is the seat of cancerous growth.

Sarcomata are not so often met with in the digestive organs. When present, they generally involve the peritoneum or the mesenteric glands. The liver is sometimes attacked, the stomach rarely.

Benign tumours are not of common occurrence in the digestive organs. Simple growths of the salivary glands, cysts of the pancreas and polypoid tumours of the rectum are the most frequent.

Animal parasites.

The intestinal canal is the habitat of the majority of animal parasites found in man. Frequently their presence leads to no morbid symptoms, local or general; nor are the symptoms, when they do arise, always characteristic of the presence of parasites alone. Discovery of their bodies, or of their eggs, in the stools is in most instances the only satisfactory proof of their presence. The parasites found in the bowel belong principally to two natural groups, Protozoa and Metazoa. The great class of the Protozoa furnish amoebae, members of Sporozoa and Infusoria. The amoebae are almost invariably found in the large intestine; one species, indeed, is termed _Amoeba coli_. The frequently observed relation between attacks of dysentery and the presence of amoebae in the stools has led to the proposition that an _Amoeba dysenterica_ exists, causing the disease--a theory supported by the detection of amoebae in the contents of dysenteric abscesses of the liver. No symptoms of injury to health appear to accompany the presence of Sporozoa in the bowel, while the species of Infusoria found in it, the _Cercomonas_, and _Trichomonas intestinalis_, and the _Balantidium coli_, may or may not be guilty of prolonging conditions within the bowel as have previously set up diarrhoea.

The Metazoa supply examples of intestinal parasites from the classes Annuloida and Nematoidea. To the former class belong the various tapeworms found in the small intestine of man. They, like other intestinal parasites, are destitute of any power of active digestion, simply absorbing the nutritious proceeds of the digestive processes of their hosts. Nematode worms infest both the small and large intestine; _Ascaris lumbricoides_, the common round worm, and the male _Oxyuris vermicularis_ are found in the small bowel, the adult female _Oxyuris vermicularis_ and the _Tricocephalus dispar_ in the large.

The eggs of the _Trichina spiralis_, when introduced with the food, develop in the bowel into larval forms which invade the tissues of the body, to find in the muscles congenial spots wherein to reach maturity. Similarly, the eggs of the Echinococcus are hatched in the bowel, and the embryos proceed to take up their abode in the tissues of the body, developing into cysts capable of growth into mature worms after their ingestion by dogs.

Vegetable parasites.

Numbers of bacterial forms habitually infest the alimentary canal. Many of them are non-pathogenic; some develop pathogenic characters only under provocation or when a suitable environment induces them to act in such a manner; others may form the _materies morbi_ of special lesions, or be casual visitors capable of originating disease if opportunity occurs. Apart from those organisms associated with acute infective diseases, disturbances of function and physical lesions may be the result of abnormal bacterial activity in the canal; and these disturbances may be both local and general. Many of the bacteria commonly present produce putrefactive changes in the contents of the tract by their metabolic processes. They render the medium they grow in alkaline, produce different gases and elaborate more or less virulent toxins. Other species set up an acid fermentation, seldom accompanied by gas or toxin formation. The products of either class are inimical to the free growth of members of the other. The species which produce acids are more resistant to the action of acids. Thus, when the contents of the stomach possess a normal or excessive proportion of free hydrochloric acid, a much larger number of putrefactive and pathogenic organisms in the food are destroyed or inhibited than of the bacteria of acid fermentation. Diminished gastric acidity allows of the entry of a greater number of putrefactive (and pathogenic) types, with, as a consequence, increased facilities for their growth and activity, and the appearance of intestinal derangements.

TABLE I.

+-----------------------------+--------------------------+--------------------------+ | Males. | Females. | Both Sexes. | +---------------------+-------+------------------+-------+------------------+-------+ |Organ or Tissue in | Per- |Organ or Tissue in| Per- |Organ or Tissue in| Per- | |Order of Frequency. |centage|Order of Frequency|centage|Order of Frequency|centage| +---------------------+-------+------------------+-------+------------------+-------+ | 1 Stomach | 22.56 | 1 Stomach | 22.37 | 1 Stomach | 22.49 | | 2 Lip | 12.94 | 2 Rectum | 17.24 | 2 Rectum | 13.12 | | 3 Rectum | 11.57 | 3 Liver | 15.50 | 3 Liver | 10.02 | | 4 Tongue | 11.36 | 4 Peritoneum | 7.86 | 4 Lip | 9.89 | | 5 Oesophagus | 10.90 | 5 Oesophagus | 5.33 | 5 Oesophagus | 9.29 | | 6 Liver | 7.80 | 6 Sigmoid | 4.53 | 6 Tongue | 8.96 | | 7 Jaw | 6.38 | 7 Pancreas | 3.52 | 7 Jaw | 5.65 | | 8 Mouth | 2.88 | 8 Tongue | 3.12 | 8 Peritoneum | 2.94 | | 9 Tonsils | 2.09 | 9 Omentum | 2.98 | 9 Sigmoid | 2.56 | |10 Sigmoid flexure | 1.77 |10 Lip | 2.57 |10 Mouth | 2.40 | |11 Parotid | 1.10 |11 Jaw | 1.97 |11 Pancreas | 1.80 | |12 Pancreas | " |12 Colon | 1.84 |12 Tonsils | 1.35 | |13 Caecum | 0.94 |13 Abdomen | " |13 Omentum | 1.25 | |14 Peritoneum | " |14 Intestine | 1.56 |14 Parotid | 1.12 | |15 Colon | 0.89 |15 Caecum | 1.37 |15 Colon | " | |16 Pharynx | |16 Mouth | 1.18 |16 Caecum | 1.08 | |17 Intestine (site | |17 Parotid | " |17 Intestine | 1.00 | | unknown) | 0.79 |18 Splenic flexure| 0.98 |18 Abdomen | " | |18 Abdomen | 0.71 |19 Jejunum and | |19 Pharynx | 0.62 | |19 Mesentery | 0.55 | ileum | 0.68 |20 Mesentery | 0.52 | |20 Omentum | " |20 Tonsils | 0.68 |21 Jejunum and | | |21 Hepatic flexure | 0.39 |21 Pharynx | 0.40 | ileum | 0.44 | |22 Submaxillary gland| 0.31 |22 Hepatic flexure| " |22 Hepatic flexure| " | |23 Jejunum and ileum | " |23 Mesentery | " |23 Splenic flexure| 0.28 | |24 Duodenum | 0.23 |24 Submaxillary | 0.20 |24 Submaxillary | 0.22 | |25 Splenic flexure | 0.15 |25 Duodenum | " |25 Duodenum | | +---------------------+-------+------------------+-------+------------------+-------+

_Note._--The figures where several organs are bracketed apply to each organ separately.

In a healthy new-born infant the mouth is free from micro-organisms, and very few are found in a breast-fed baby, but _Bacillus lactis_ may be found where the child is bottle fed. If there is trouble with the first dentition and food is allowed to collect, staphylococci, streptococci, pneumococci and colon bacilli may be present. Even in healthy babies _Oidium albicans_ may be present, and in older children the pseudo-diphtheria bacillus. From carious teeth may be isolated streptothrix, leptothrix, spirilla and fusiform bacilli. Under conditions of health these micro-organisms live in the mouth as saprophytes, and show no virulence when cultivated and injected into animals. The two common pyogenetic organisms, _Staphylococcus albus_ and _brevis_, show no virulence. Also the pneumococcus, though often present, must be raised in virulence before it can produce untoward results. The foulness of the mouth is supposed to be due to the colon bacillus and its allies, but those obtained from the mouth are innocuous. Also to enable the _Oidium albicans_ to attack the mucous membrane there must be some slight inflammation or injury. The micro-organisms found in the stomach gain access to that organ in the food or by regurgitation from the small intestine. Most are relatively inert, but some have a special fermentative action on the food (see NUTRITION). Abelous isolated sixteen distinct species of organism from a healthy stomach, including Sarcinae, _B. lactis_, _pyocyaneus_, _subtilis_, _lactis erythrogenes_, _amylobacter_, _megatherium_, and _Vibrio rugula_.

Physical abnormalities

Hare-lip, cleft palate, hernia and imperforate anus are physical abnormalities which are interesting to the surgeon rather than to the pathologist. The oesophagus may be the seat of a diverticulum, or blind pouch, usually situated in its lower half, which in most instances is probably partly acquired and partly congenital; a local weakness succumbing to pressure. Hypertrophy of the muscular coat of the pyloric region is an infrequent congenital gastric anomaly in infants, preventing the passage of food into the bowel, and causing death in a short time. Incomplete closure of the vitelline duct results in the presence of a diverticulum--Meckel's--generally connected with the ileum, mainly important by reason of the readiness with which it occasions intestinal obstruction. Idiopathic congenital dilatation of the colon has been described.

TABLE II.

+------------------+--------+------------------+--------+------------------+--------+ | Males. | Per- | Females. | Per- | Total. | Per- | | |centage.| |centage.| |centage.| +------------------+--------+------------------+--------+------------------+--------+ | 1 Mouth and | | 1 Intestines | 28.9 | 1 Oesophagus and | | | pharynx | 37.85 | 2 Oesophagus and | | stomach | 31.78 | | 2 Oesophagus and | | stomach | 27.7 | 2 Mouth and | | | stomach | 33.46 | 3 Liver | 15.5 | pharynx | 30.27 | | 3 Intestines | 17.04 | 4 Peritoneum | 13.1 | 3 Intestines | 20.42 | | 4 Liver | 7.8 | 5 Mouth and | | 4 Liver | 10.02 | | 5 Peritoneum | 2.75 | pharynx | 11.3 | 5 Peritoneum | 5.71 | | 6 Pancreas | 1.1 | 6 Pancreas | 3.5 | 6 Pancreas | 1.80 | +------------------+--------+------------------+--------+------------------+--------+

Traction diverticula of the oesophagus not uncommonly occur as sequels to suppurative inflammation of cervical lymphatic glands. More frequently dilatation of a section is met with, due as a rule to the presence of a stricture. The stomach often diverges from the normal in size, shape and position. Normally capable in the adult of containing from fifty to sixty ounces, either by reason of organic disease, or as the result of functional disturbance, its capacity may vary enormously. The writer has seen post mortem a stomach which held a gallon (160 ounces), and again one holding only two ounces. Cancer spread over a large area and cirrhosis of the stomach wall cause diminution in capacity; pyloric obstruction, weakness of the muscular coat, and nervous influences are associated with dilatation. A peculiar distortion of the shape of the stomach follows cicatrization of ulcers of greater or lesser curvature; the gastric cavity becomes "hour-glass" in shape. In addition, the stomach may be displaced downwards as a whole, a condition known as gastroptosis: if the pyloric portion only be displaced, the lesion is termed pyloroptosis. Ptoses of other abdominal organs are described; the liver, transverse colon, spleen and kidneys may be involved. Displacements downwards of the stomach and transverse colon, along with a movable right kidney and associated with dyspepsia and neurasthenia, form the malady termed by Glénard enteroptosis. A general visceroptosis often occurs in those patients who have some tuberculous lesion of the lungs or elsewhere, this disease causing a general weakening and subsequent stretching of all ligaments. Displacements of the abdominal viscera are almost invariably accompanied by symptoms of dyspepsia of a neurotic type. The rectum is liable to prolapse, consequent upon constipation and straining at stool, or following local injuries of the perineal floor.

Influence of the nervous system.

Every pathological lesion shown by digestive organs is closely associated with the state of the nervous system, general or local; so stoppage of active gastric digestive processes after profound nervous shock, and occurrence of nervous diarrhoea from the same cause. Gastric dyspepsia of nervous origin presents most varied and contradictory symptoms: diminished acidity of the gastric juice, hyper-acidity, over-production, arrest of secretion, lessened or increased movements, greater sensitiveness to the presence of contents, dilatation or spasm. Often the nervous cause can be traced back farther,--in females, frequently to the pelvic organs; in both sexes, to the condition of the blood, the brain or the bowel. Unhealthy conditions related to evacuation of the bowel-contents commonly induce reflex nervous manifestations of abnormal character referred to the stomach and liver. Gastric disturbances similarly react upon the proper conduct of intestinal functions.

_Local Diseases._

_The Mouth._--The lining membrane of the cheeks inside the mouth, of the gums and the under-surface and edges of the tongue, is often the seat of small irritable ulcers, usually associated with some digestive derangement. A crop of minute vesicles known as Koplik's spots over these parts has been lately stated by Koplik to be an early symptom of measles. Xerostomia, or dry mouth, is a rare condition, connected with lack of salivary secretion. Gangrenous stomatitis, cancrum oris, or noma, occasionally attacks debilitated children, or patients convalescing from acute fevers, more especially after measles. It commences in the gums or cheeks, and causes widespread sloughing of the adjacent soft parts--it may be of the bones.

_The Stomach._--It were futile to attempt to enumerate all the protean manifestations of disturbance which proceed from a disordered stomach. The possible permutations and combinations of the causes of gastric vagaries almost reach infinity. Idiosyncrasy, past and present gastric education, penury or plethora, actual digestive power, motility, bodily requirements and conditions, environment, mental influences, local or adjacent organic lesions, and, not least, reflex impressions from other organs, all contribute to the variance.

Ulcer of the stomach, however--the perforating gastric ulcer--occupies a unique position among diseases of this organ. Gastric ulcers are circumscribed, punched out, rarely larger than a sixpenny-bit, funnel-shaped, the narrower end towards the peritoneal coat, and distributed in those regions of the stomach wall which are most exposed to the action of the gastric contents. They occur most frequently in females, especially if anaemic, and are usually accompanied by excess of acid, actual or relative to the state of the blood, in the stomach contents. Local pain, dorsal pain, generally to the left of the eighth or ninth dorsal spinous process, and haematernesis and melaena, are symptomatic of it. The amount of blood lost varies with the rapidity of ulcer formation and the size of vessel opened into. Fatal results arise from ulceration into large blood-vessels, followed by copious haemorrhage, or by perforation of the ulcer into the peritoneal cavity. Scars of such ulcers may be found post mortem, although no symptoms of gastric disease have been exhibited during life; gastric ulcers, therefore, may be latent.

Irritation of the sensory nerve-endings in the stomach wall from the presence of an increased proportion of acid, organic or mineral, in the stomach contents is accountable for the well known symptom heartburn. Water-brash is a term applied to eructation of a colourless, almost tasteless fluid, probably saliva, which has collected in the lower part of the oesophagus from failure of the cardiac sphincter of the stomach to relax; reversed oesophageal peristalsis causing regurgitation. A similar reversed action serves in merycism, or rumination, occasionally found in man, to raise part of the food, lately ingested, from the stomach to the mouth. Vomiting also is aided by reversed peristaltic action, both of the stomach and the oesophagus, with the help of the diaphragm and the muscles of the anterior abdominal wall. Emesis may be caused both by local nervous influence, and through the central nervous mechanism either reflexly or from the direct action of substances circulating in the blood. Further, the causal agent acting on the central nervous apparatus may be organic or functional, as well as medicinal. Vomiting without any apparent cause suggests nervous lesions, organic or reflex. The obstinate vomiting of pregnancy is a case in point. Here the primary cause proceeds reflexly from the pelvis. In females the pelvic organs are often the true source of emesis. Haematemesis accompanies gastric ulcer, cancer, chronic congestion with haemorrhagic erosion, congestion of the liver, or may follow violent acts of vomiting. In cases of ulcer the blood is usually bright and in considerable amount; in cancer, darker, like coffee-grounds; and in cases of erosion, in smaller quantity and of bright colour. The reaction of the stomach contents, if the cause be doubtful, yields valuable aid towards a diagnosis. Of increased acidity in gastric ulcer, normal in hepatic congestion, it is diminished in cancer; but as the acid present in cancer is largely lactic, analysis of the gastric contents must often be a _sine qua non_, because hyperacidity from lactic may obscure hypoacidity of hydrochloric acid.

Flatulence usually results from fermentative processes in the stomach and bowel, as the outcome of bacterial activity. A different form of flatulence is common in neurotic individuals: in such the gas evolved consists simply in carbonic acid liberated from the blood, and its evolution is generally characterized by rapid development and by lack of all fermentative signs.

_The Liver._--The liver is an organ frequently libelled for the delinquencies of other organs, and regarded as a common source of ill. In catarrhal jaundice it is in most cases the bowel that is at fault, the liver acting properly, but unable to get rid of all the bile produced. The liver suffers, however, from several diseases of its own. Its fibrous or connective tissue is very apt to increase at the expense of the cellular elements, destroying their functions. This cirrhotic process usually follows long-continued irritation, such as is produced by too much alcohol absorbed from the bowel habitually, the organ gradually becoming harder in texture and smaller in bulk. Hypertrophic cirrhosis of the liver is not uncommonly met with, in which the liver is much increased in size, the "unilobular" form, also of alcoholic origin. In still-born children and in some infants a form of hypertrophic cirrhosis is occasionally seen, probably of hereditary syphilitic origin. Acute congestion of the liver forms an important symptom of malarial fever, and often leads in time to establishment of cirrhotic changes; here the liver is generally enlarged, but not invariably so, and the part played by alcohol in its causation has still to be investigated. Acute yellow atrophy of the liver is a disease _sui generis_. Of rare occurrence, possibly of toxic origin, it is marked by jaundice, at first of usual type, later becoming most intense; by vomiting; haemorrhages widely distributed; rapid diminution in the size of the liver; the appearance of leucin and tyrosin in the urine, with lessened urea; and in two or three days, death. The liver after death is soft, of a reddish colour dotted with yellow patches, and weighs only about a third part of the normal--about 1½ lb in place of 3¾ lb. A closely analogous affection of the liver, known as Weil's disease, is of infectious type, and has been noted in epidemic form. In this the spleen and liver are commonly but not always swollen, and the liver is often tender on pressure. As a large proportion of the sufferers from this disease have been butchers, and the epidemics have occurred in the hot season of the year, it probably arises from contact with decomposing animal matter. Hepatic abscess may follow on an attack of amoebic dysentery, and is produced either by infection through the portal vein, or by direct infection from the adjacent colon. In general pyaemia multiple small abscesses may occur in the liver.

_The Gall-Bladder._--The formation of biliary calculi in the gall-bladder is the chief point of interest here. At least 75% of such cases occur in women, especially in those who have borne children. Tight-lacing has been stated to act as an exciting cause, owing to the consequent retardation of the flow of bile. Gall-stones may number from one to many thousands. They are largely composed of cholesterin, combined with small amounts of bile-pigments and acids, lime and magnesium salts. Their presence may give rise to no symptoms, or may cause violent biliary colic, and, if the bile-stream be obstructed, to jaundice. Inflammatory processes may be initiated in the gall-bladder or the bile-ducts, catarrhal or suppurative in character.

_The Pancreas._--Haemorrhages into the body of the pancreas, acute and chronic inflammation, calculi, cysts and tumours, among which cancer is by far the most common, are recognized as occurring in this organ; the point of greatest interest regarding them lies in the relations established between pancreatic disease and diabetes mellitus, affections of the gland frequently being complicated by, and probably causing, the appearance of sugar in the urine.

_The Small Intestine._--Little remains to be added to the account of inflammatory lesions in connexion with the small intestine. It offers but few conditions peculiar to itself, save in typhoid fever, and the ease with which it contrives to become kinked, or intussuscepted, producing obstruction, or to take part in hernial protrusions. The first section, the duodenum, is subject to development of ulcers very similar to those of the gastric mucous membrane. For long duodenal ulceration has been regarded as a complication of extensive burns of the skin, but the relationship between them has not yet been quite satisfactorily explained. The condition of colic in the bowel usually arises from overdistension of some part of the small gut with gas, the frequent sharp turns of the gut facilitating temporary closure of its lumen by pressure of the dilated gut near a curve against the part beyond. In the large bowel accumulations of gas seldom cause such acute symptoms, having a readier exit.

_The Large Intestine._--The colon, especially the ascending portion, may become immensely dilated, usually after prolonged constipation and paralysis of the gut; occasionally the condition is congenital. Straining efforts made in defaecation may often account for prolapse of the lower end of the rectum through the anus. Haemorrhage from the bowel is usually a sign of disease situated in the large intestine: if bright in colour, the source is probably low down; if dark, from the caecum or from above the ileo-caecal valve. Blood after a short stay in any section of the alimentary canal darkens, and eventually becomes almost black in colour. (A. L. G.; M. F.*)

DIGGES, WEST (1720-1786), English actor, made his first stage appearance in Dublin in 1749 as Jaffier in _Venice Preserved_; and both there and in Edinburgh until 1764 he acted in many tragic rôles with success. He was the original "young Norval" in Home's _Douglas_ (1756). His first London appearance was as Cato in the Haymarket in 1777, and he afterwards played Lear, Macbeth, Shylock and Wolsey. In 1881 he returned to Dublin and retired in 1784.

DIGIT (Lat. _digitus_, finger), literally a finger or toe, and so used to mean, from counting on the fingers, a single numeral, or, from measuring, a finger's breadth. In astronomy a digit is the twelfth part of the diameter of the sun or moon; it is used to express the magnitude of an eclipse.

DIGITALIS. The leaves of the foxglove (q.v.), gathered from wild plants when about two-thirds of their flowers are expanded, deprived usually of the petiole and the thicker part of the midrib, bitter taste; and to preserve their properties they must be kept excluded from light in stoppered bottles. They are occasionally adulterated with the leaves of _Inula Conyza_, ploughman's spikenard, which may be distinguished by their greater roughness, their less divided margins, and their odour when rubbed; also with the leaves of _Symphytum officinale_, comfrey, and of _Verbascum Thapsus_, great mullein, which unlike those of the foxglove have woolly upper and under surfaces. The earliest known descriptions of the foxglove are those given by Leonhard Fuchs and Tragus about the middle of the 16th century, but its virtues were doubtless known to herbalists at a much remoter period. J. Gerarde, in his _Herbal_ (1597), advocates the use of foxglove for a variety of complaints; and John Parkinson, in the _Theatrum Botanicum_, or _Theater of Plants_ (1640), and later W. Salmon, in _The New London Dispensatory_, similarly praised the remedy. Digitalis was first brought prominently under the notice of the medical profession by Dr W. Withering, who, in his _Account of the Foxglove_ (1785), gave details of upwards of 200 cases chiefly dropsical, in which it was used.

Digitalis contains four important glucosides, of which three are cardiac stimulants. The most powerful is _digitoxin_ C34H54O11, an extremely poisonous and cumulative drug, insoluble in water. _Digitalin_, C35H56O14, is crystalline and is also insoluble in water. _Digitalein_ is amorphous but readily soluble in water. It can therefore be administered subcutaneously, in doses of about one-hundredth of a grain. _Digitonin_, on the other hand, is a cardiac depressant, and has been found to be identical with saponin, the chief constituent of senega root. There are numerous preparations, patent and pharmacopeial, their composition being extremely varied, so that, unless one has reason to be certain of any particular preparation, it is almost better to use only the dried leaves themselves in the form of a powder (dose ½-2 grains). The pharmacopeial tincture may be given in doses of five to fifteen minims, and the infusion has the unusually small dose of two to four drachms--the dose of other infusions being an ounce or more. The tincture contains a fair proportion of both digitalin and digitoxin.

Digitalis leaves have no definite external action. Taken by the mouth, the drug is apt to cause considerable digestive disturbance, varying in different cases and sometimes so severe as to cause serious difficulty. This action is probably due to the digitonin, which is thus a constituent in every way undesirable. The all-important property of the drug is its action on the circulation. Its first action on any of the body-tissues is upon unstriped muscle, so that the first consequence of its absorption is a contraction of the arteries and arterioles. No other known drug has an equally marked action in contracting the arterioles. As the vaso-motor centre in the medulla oblongata is also stimulated, as well as the contractions of the heart, there is thus trebly caused a very great rise in the blood-pressure.

The clinical influence of digitalis upon the heart is very well defined. After the taking of a moderate dose the pulse is markedly slowed. This is due to a very definite influence upon the different portions of the cardiac cycle. The systole is not altered in length, but the diastole is very much prolonged, and since this is the period not only of cardiac rest but also of cardiac "feeding"--the coronary vessels being compressed and occluded during systole--the result is greatly to benefit the nutrition of the cardiac muscle. So definite is this that, despite a great increase in the force of the contractions and despite experimental proof that the heart does more work in a given time under the influence of digitalis, the organ subsequently displays all the signs of having rested, its improved vigour being really due to its obtaining a larger supply of the nutrient blood. Almost equally striking is the fact that digitalis causes an irregular pulse to become regular. Added to the greater force of cardiac contraction is a permanent tonic contraction of the organ, so that its internal capacity is reduced. The bearing of this fact on cases of cardiac dilatation is evident. In larger doses a remarkable sequel to these actions may be observed. The cardiac contractions become irregular, the ventricle assumes curious shapes--"hour-glass," &c.--becomes very pale and bloodless, and finally the heart stops in a state of spasm, which shortly afterwards becomes rigor-mortis. Before this final change the heart may be started again by the application of a soluble potassium salt, or by raising the fluid pressure within it. Clinically it is to be observed that the drug is cumulative, being very slowly excreted, and that after it has been taken for some time the pulse may become irregular, the blood-pressure low, and the cardiac pulsations rapid and feeble. These symptoms with more or less gastro-intestinal irritation and decrease in the quantity of urine passed indicate digitalis poisoning. The initial action of digitalis is a stimulation of the cardiac terminals of the vagus nerves, so that the heart's action is slowed. Thereafter follows the most important effect of the drug, which is a direct stimulation of the cardiac muscle. This can be proved to occur in a heart so embryonic that no nerves can be recognized in it, and in portions of cardiac muscle that contain neither nervecells nor nerve-fibres.

The action of this drug on the kidney is of importance only second to its action on the circulation. In small or moderate doses it is a powerful diuretic. Though Heidenhain asserts that rise in the renal blood-pressure has not a diuretic action per se, it seems probable that this influence of the drug is due to a rise in the general blood-pressure associated with a relatively dilated condition of the renal vessels. In large doses, on the other hand, the renal vessels also are constricted and the amount of urine falls. It is probable that digitalis increases the amount of water rather than that of the urinary solids. In large doses the action of digitalis on the circulation causes various cerebral symptoms, such as seeing all objects blue, and various other disturbances of the special senses. There appears also to be a specific action of lowering the reflex excitability of the spinal cord.

Digitalis is used in therapeutics exclusively for its action on the circulation. In prescribing this drug it must be remembered that fully three days elapse before it gets into the system, and thus it must always be combined with other remedies to tide the patient over this period. It must never be prescribed in large doses to begin with, as some patients are quite unable to take it, intractable vomiting being caused. The three days that must pass before any clinical effect is obtained renders it useless in an emergency. A certain consequence of its use is to cause or increase cardiac hypertrophy--a condition which has its own dangers and ultimately disastrous consequences, and must never be provoked beyond the positive needs of the case. But digitalis is indicated whenever the heart shows itself unequal to the work it has to perform. This formula includes the vast majority of cardiac cases. The drug is contra-indicated in all cases where the heart is already beating too slowly; in aortic incompetence--where the prolongation of diastole increases the amount of the blood that regurgitates through the incompetent valve; in chronic Bright's disease and in fatty degeneration of the heart--since nothing can cause fat to become contractile.

DIGNE, the chief town of the department of the Basses Alpes, in S.E. France, 14 m. by a branch line from the main railway line between Grenoble and Avignon. Pop. (1906), town, 4628; commune, 7456. The Ville Haute is built on a mountain spur running down to the left bank of the Bléone river, and is composed of a labyrinth of narrow winding streets, above which towers the present cathedral church, dating from the end of the 15th century, but largely reconstructed in modern times, and the former bishop's palace (now the prison). The fine Boulevard Gassendi separates the Ville Haute from the Ville Basse, which is of modern date. The old cathedral (Notre Dame du Bourg) is a building of the 13th century, but is now disused except for funerals: it stands at the east end of the Ville Basse. The neighbourhood of Digne is rich in orchards, which have long made the town famous in France for its preserved fruits and confections. It is the _Dinia_ of the Romans, and was the capital of the Bodiontii. From the early 6th century at least it has been an episcopal see, which till 1790 was in the ecclesiastical province of Embrun, but since 1802 in that of Aix en Provence. The history of Digne in the middle ages is bound up with that of its bishops, under whom it prospered greatly. But it suffered much during the religious wars of the 16th and 17th centuries, when it was sacked several times. A little way off, above the right bank of the Bléone, is Champtercier, the birthplace of the astronomer Gassendi (1592-1655), whose name has been given to the principal thoroughfare of the little town.

See F. Guichard, _Souvenirs historiques sur la ville de Digne et ses environs_ (Digne, 1847). (W. A. B. C.)

DIGOIN, a town of east-central France, in the department of Saône-et-Loire, on the right bank of the Loire, 55 m. W.N.W. of Mâcon on the Paris-Lyon railway. Pop. (1906) 5321. It is situated at the meeting places of the Loire, the Lateral canal of the Loire and the Canal du Centre, which here crosses the Loire by a fine aqueduct. The town carries on considerable manufactures of faience, pottery and porcelain. The port on the Canal du Centre has considerable traffic in timber, sand, iron, coal and stone.

DIJON, a town of eastern France, capital of the department of Côte d'Or and formerly capital of the province of Burgundy, 195 m. S.E. of Paris on the Paris-Lyon railway. Pop. (1906) 65,516. It is situated on the western border of the fertile plain of Burgundy, at the foot of Mont Afrique, the north-eastern summit of the Côte d'Or range, and at the confluence of the Ouche and the Suzon; it also has a port on the canal of Burgundy. The great strategic importance of Dijon as a centre of railways and roads, and its position with reference to an invasion of France from the Rhine, have led to the creation of a fortress forming part of the Langres group. There is no _enceinte_, but on the east side detached forts, 3 to 4 m. distant from the centre, command all the great roads, while the hilly ground to the west is protected by Fort Hauteville to the N.W. and the "groups" of Motte Giron and Mont Afrique to the S.W., these latter being very formidable works. Including a fort near Saussy (about 8 m. to the N.W.) protecting the water-supply of Dijon, there are eight forts, besides the groups above mentioned. The fortifications which partly surrounded the old and central portion of the city have disappeared to make way for tree-lined boulevards with fine squares at intervals. The old churches and historic buildings of Dijon are to be found in the irregular streets of the old town, but industrial and commercial activity has been transferred to the new quarters beyond its limits. A fine park more than 80 acres in extent lies to the south of the city, which is rich in open spaces and promenades, the latter including the botanical garden and the Promenade de l'Arquebuse, in which there is a black poplar famous for its size and age.

The cathedral of St Bénigne, originally an abbey church, was built in the latter half of the 13th century on the site of a Romanesque basilica, of which the crypt remains. The west front is flanked by two towers and the crossing is surmounted by a slender timber spire. The plan consists of three naves, short transepts and a small choir, without ambulatory, terminating in three apses. In the interior there is a fine organ and a quantity of statuary, and the vaults contain the remains of Philip the Bold, duke of Burgundy, and Anne of Burgundy, daughter of John the Fearless. The site of the abbey buildings is occupied by the bishop's palace and an ecclesiastical seminary. The church of Notre-Dame, typical of the Gothic style of Burgundy, was erected from 1252 to 1334, and is distinguished for the grace of its interior and the beauty of the western façade. The portal consists of three arched openings, above which are two stages of arcades, open to the light and supported on slender columns. A row of gargoyles surmounts each storey of the façade, which is also ornamented by sculptured friezes. A turret to the right of the portal carries a clock called the Jaquemart, on which the hours are struck by two figures. The church of St Michel belongs to the 15th century. The west façade, the most remarkable feature of the church, is, however, of the Renaissance period. The vaulting of the three portals is of exceptional depth owing to the projection of the lower storey of the façade. Above this storey rise two towers of five stages, the fifth stage being formed by an octagonal cupola. The columns decorating the façade represent all the four orders. The design of this façade is wrongly attributed to Hugues Sambin (fl.