the cosine of the angle between the direction of the force and the line 00, produced, so that

k = ll' + mm' + nn'.

Consider at present the first term of the right-hand side of (29). Since the radius vector drawn from 0 to any element of T ultimately coincides with 00,, we may put / outside the integral signs, and replace do by rdS. Moreover, since this term vanishes except when at lies between the greatest and least values of the radius vector drawn from 0 to any element of T, we may replace t outside the integral signs by r/a. Conceive a series of spheres, with radii ar, 2aт...naт,... described round O, and let the nth of these be the first which cuts T. Let S1, S... be the areas of the surfaces of the spheres, beginning with the nth, which lie within T; then

2

...

SS(9%)ardS = kтF (t − nт) S ̧ + kтF {t − (n + 1) 7} S1⁄2 + But F(t- nr), F{t-(n+1) 7}... are ultimately equal to each other, and to

and aTS,+ aTS, + ... is ultimately equal to T. Hence we get, for the part of which arises from the first of the double integrals,

The second of the double integrals is to be treated in exactly the same way.

To find what the triple integral becomes, let us consider first only the impulse which was communicated at the beginning of the time t―nt, where nT lies between the limits r/a and r/b, and is not so nearly equal to one of these limits that any portion of the space T lies beyond the limits of integration. Then we must write nʊ for t in the coefficient, and 3lq, -u, becomes ultimately (3lk — l') τ F' (t — nт), and, as well as r, is ultimately constant in the triple integration. Hence the triple integral ultimately becomes

and we have now to perform a summation with reference to

different values of n, which in the limit becomes an integration. Putting nTt', we have ultimately

7 = dí', Σnt . 7 F (t − n7) = ['" ť F (t − t) dt.

It is easily seen that the terms arising from the triple integral when it has to be extended over a part only of the space T vanish in the limit. Hence we have, collecting all the terms, and expressing F (t) in terms of ƒ (t),

η

To get ŋ and , we have only to pass from 7, l' to m, m' and then to n, n'. If we take 00, for the axis of x, and the plane passing through 00, and the direction of the force for the plane xz, and put a for the inclination of the direction of the force to 00, produced, we shall have

1

whence

l = 1, m = 0, n = 0, l' = k = cos z, m' = 0, n' = sin 2 ;

In the investigation, it has been supposed that the force began to act at the time 0, before which the fluid was at rest, so that f(t)=0 when t is negative. But it is evident that exactly the same reasoning would have applied had the force begun to act at any past epoch, as remote as we please, so that we are not obliged to suppose f (t) equal to zero when t is negative, and we may even suppose f(t) periodic, so as to have finite values from t∞ to

t = +∞.

By means of the formula (39), it would be very easy to write down the expressions for the disturbance due to a system of forces acting throughout any finite portion of the medium, the disturbing

force varying in any given manner, both as to magnitude and direction, from one point of the medium to another, as well as from one instant of time to another.

1

The first term in § represents a disturbance which is propagated from 0, with a velocity a. Since there is no corresponding term in ʼn or, the displacement, as far as relates to this disturbance, is strictly normal to the front of the wave. The first term in represents a disturbance which is propagated from 0, with a velocity b, and as far as relates to this disturbance the displacement takes place strictly in the front of the wave. The remaining terms in § and represent a disturbance of the same kind as that which takes place in an incompressible fluid in consequence of the motion of solid bodies in it. If f (t) represent a force which acts for a short time, and then ceases, ƒ (t −t') will differ from zero only between certain narrow limits of t, and the integral contained in the last terms of and will be of the order r, and therefore the terms themselves will be of the order r2, whereas the leading terms are of the order r1. Hence in this case the former terms will not be sensible beyond the immediate neighbourhood of O̟1. The same will be true if ƒ (t) represent a periodic force, the mean value of which is zero. But if f (t) represent a force always acting one way, as for example a constant force, the last terms in & and will be of the same order, when r is large, as the first terms.

28. It has been remarked in the introduction that there is strong reason for believing that in the case of the luminiferous ether the ratio of a to b is extremely large, if not infinite. Consequently the first term in §, which relates to normal vibrations, will be insensible, if not absolutely evanescent. In fact, if the ratio of a to b were no greater than 100, the denominator in this term. would be 10000 times as great as the denominator of the first term in C. Now the molecules of a solid or gas in the act of combustion are probably thrown into a state of violent vibration, and may be regarded, at least very approximately, as centres of disturbing forces. We may thus see why transversal vibrations should alone be produced, unaccompanied by normal vibrations, or at least by any which are of sufficient magnitude to be sensible. If we could be sure that the ether was strictly incompressible, we should of course be justified in asserting that normal vibrations are impossible.

29. If we suppose a = ∞, and ƒ(t) = c sin 2πbt/λ, we shall get

(41);

and we see that the most important term in § is of the order λ/πr compared with the leading term in , which represents the transversal vibrations properly so called. Hence &, and the second and third terms in , will be insensible, except at a distance from 0, comparable with λ, and may be neglected; but the existence of terms of this nature, in the case of a spherical wave whose radius is not regarded as infinite, must be borne in mind, in order to understand in what manner transversal vibrations are compatible with the absence of dilatation or condensation.

30. The integration of equations (18) might have been effected somewhat differently by first decomposing the given functions §, no, 。, and u, v, w, into two parts, as in Art. 8, and then treating each part separately. We should thus be led to consider separately that part of the initial disturbance which relates to a wave of dilatation and that part which relates to a wave of distortion. Either of these parts, taken separately, represents a disturbance which is not confined to the space T, but extends indefinitely around it. Outside T, the two disturbances are equal in magnitude and opposite in sign.

SECTION III.

Determination of the Law of the Disturbance in a Secondary Wave of Light.

31. Conceive a series of plane waves of plane-polarized light propagated in vacuum in a direction perpendicular to a fixed mathematical plane P. According to the undulatory theory of light, as commonly received, that is, including the doctrine of

transverse vibrations, the light in the case above supposed consists in the vibrations of an elastic medium or ether, the vibrations being such that the ether moves in sheets, in a direction perpendicular to that of propagation, and the vibration of each particle being symmetrical with respect to the plane of polarization, and therefore rectilinear, and either parallel or perpendicular to that plane. In order to account for the propagation of such vibrations, it is necessary to suppose the existence of a tangential force, or tangential pressure, called into play by the continuous sliding of the sheets one over another, and proportional to the amount of the displacement of sliding. There is no occasion to enter into any speculation as to the cause of this tangential force, nor to entertain the question whether the luminiferous ether consists of distinct molecules or is mathematically continuous, just as there is no occasion to speculate as to the cause of gravity in calculating the motions of the planets. But we are absolutely obliged to suppose the existence of such a force, unless we are prepared to throw overboard the theory of transversal vibrations, as usually received, notwithstanding the multitude of curious, and otherwise apparently inexplicable phenomena which that theory explains with the utmost simplicity. Consequently we are led to treat the ether as an elastic solid so far as the motions which constitute light are concerned. It does not at all follow that the ether is to be regarded as an elastic solid when large displacements are considered, such as we may conceive produced by the earth and planets, and solid bodies in general, moving through it. The mathematical theories of fluids and of elastic solids are founded, or at least may be founded, on the consideration of internal pressures. In the case of a fluid, these pressures are supposed normal to the common surface of the two portions whose mutual action is considered: this supposition forms in fact the mathematical definition of a fluid. In the case of an elastic solid, the pressures are in general oblique, and may even in certain directions be wholly tangential. The treatment of the question by means of pressures presupposes the absence of any sensible direct mutual action of two portions of the medium which are separated by a small but sensible interval. The state of constraint or of motion of any element affects the pressures in the surrounding medium, and in this way one element exerts an indirect action on another from which it is separated by a sensible interval.