mutual action between any two particles is along the line joining them; but inasmuch as the law of force is entirely a matter of speculation, Green discarded the hypothesis of mutually attracting particles, and based his theory upon the assumption that;—In whatever way the elements of any material system may act upon one another, if all the internal forces be multiplied by the elements of their respective distances, the total sum for any assigned portion of the medium will be an exact differential of some function. This function is what is now known as the potential energy of the portion of the medium considered; and Green showed that in its most general form, it is a homogeneous quadratic function of what, in the language of the Theory of Elasticity, are called the six components of strain, and therefore contains twenty-one terms, whose coefficients are constant quantities. For a medium which is symmetrical with respect to three rectangular planes, the expression for the potential energy involves nine independent constants; whilst for an isotropic medium it involves only two; one of which depends upon the resistance which the medium offers to compression, or change of volume unaccompanied by change of shape, whilst the other depends upon the resistance which the medium offers to distortion or shearing stress, unaccompanied by change of volume.

184. The general theory of media, which are capable of resisting compression and distortion, is given in treatises on Elasticity; and it will therefore be unnecessary to reproduce investigations, which are to be found in such works. There are however one or two points, which require consideration; and we shall commence by examining the stresses, which act upon an element of such a medium.

EQUATIONS OF MOTION OF THE ETHER.

191

Let the figure represent a small parallelopiped of the medium. The stresses which act on the face AD are,

(i) A normal traction X parallel to Ox;

x

(ii) A tangential stress or shear Y parallel to Oy;

(iii) A tangential stress or shear Zz parallel to Oz.

Similarly the stresses which act upon the faces BD and CD, are Y, Z, X, and Z2, X2, Yz.

These are the stresses exerted on the faces AD, BD, CD of the element by the surrounding medium; the stresses exerted by the medium on the three opposite faces will be in the opposite directions.

185. In order to find the equations of motion, let u, v, w be the displacements parallel to the axes, of any point x, y, z; p the density, and X, Y, Z the components of the impressed forces per unit of mass. Then resolving parallel to the axes, we obtain

These equations express the fact, that the rates of change of the components of the linear momentum of an element of the medium, are equal to the components of the forces which act upon the element. It is however also necessary, that the rates of change of the components of the angular momentum of the element about the axes, should be equal to the components of the couples which act upon the element. Whence taking moments about the axis of a, we obtain

dow

dta

-2

de) dx dydz = [[] p (yZ – zY) dxdydz

{y (lZx+mZy + nZ2) − z (lYx + m Yy + nY2)} dS...(2),

where dS is an element of the surface of the portion of the medium considered, and l, m, n are the direction cosines of the normal at dS.

Transforming the surface integral into a volume integral', and taking account of (1); (2) reduces to

SSS (Zy - Y2) dx dydz = 0

=

(3),

which requires that ZyY2. It can similarly be shown, that X2=Z, and Y = Xy. These results show, that the component stresses are completely specified by the six quantities Xx, Yy, Z2, Yz, Zx, Xy, which we shall denote by the letters P, Q, R, S, T, U. Equations (1) may now be written

186. It is important to notice, that the fundamental principles of Dynamics require that the relations

Zy= Y2, X2 = Zx, Yx=Xy

.(5),

should exist between these stresses, because certain theories have been proposed, involving assumptions respecting the mutual reaction of ether and matter, in which these conditions are not satisfied. If however the medium were supposed to possess gyrostatic momentum, the above conditions would not be fulfilled. The conception of a medium, possessing a distribution of gyrostats (or fly-wheels), by means of which it is endowed with angular momentum, is originally due to Sir W. Thomson', and suggests a means of explaining the rotatory effects of magnetism on light.

187. We are now prepared to apply the Theory of Elasticity to isotropic media, and we shall adopt the notation of Thomson and Tait's Natural Philosophy, for stresses, strains and elastic constants; so that k denotes the resistance to compression, n the rigidity, and we shall often employ m to denote k+žn.

1 The transformation may be effected by the theorem proved in my treatise on Hydrodynamics, vol. 1. § 7.

2 Proc. Lond. Math. Soc. vol. vI. p. 190; Larmor, Ibid. vol. xxI. p. 423; and vol. XXIII.

It is also convenient to denote the dilatation by 8, and the rotations by έ, n,, where these quantities are defined by the

The rotations, n, are quantities analogous to the components of molecular rotation in Hydrodynamics.

The potential energy W of a homogeneous isotropic elastic medium is given in terms of the strains by the equation'

W = { (m + n) S2 + √n {a2 + b2 + c2 − 4 (ef + fg + ge)}...(8). The work done by the stresses P, Q, &c. in producing the infinitesimal strains de, df, &c. is

1 It is interesting to notice, that (8) is an example of an application of the Theory of Invariants to Physics. The three invariants, which involve the first and second powers of differential coefficients of the first order of u, v, w with respect to x, y, z are

e+f+g, a2+b2+c2 - 4 (ef +fg+ge), and 2+n2 +5o.

Since the value of W must be independent of the directions of the axes, and must also be a quadratic function, its most general form is

482+B {a2+b+c2-4 (ef+fg+ge)}+C (+n2+5o).

Equations (5) require that C=0; whence the expression reduces to (8).

B. O.

13

If the medium is isotropic, and no bodily forces act, the equations of motion are obtained by substituting the values of the stresses from (11) in (4), and taking account of (6); we thus obtain the following equations of motion in terms of the displacements, viz.

These are the equations of motion of the ether according to Green's Theory.

188. If we differentiate (12) with respect to x, y, z, add and take account of (7), we obtain

If we eliminate ♪ between each pair of equations (12) and take

It therefore follows, that the waves which can be propagated in the medium consist of two distinct types; one of which involves change of volume without rotation, and whose velocity of propagation is seen from (13) to be equal to (m + n)/pt; whilst the other involves rotation and distortion, without change of volume, and whose velocity of propagation is (n/p). The first type of waves depends partly on the rigidity and partly on the elasticity of volume; whilst the second type depends solely on the rigidity, and is therefore incapable of being propagated in a medium devoid of rigidity, such as a perfect gas. Hence if any disturbance, which involves change of volume and distortion, be communicated to a portion of the medium, two distinct trains of waves will be produced; one of which consists of a condensation