Sir George Gabriel Stokes

The researches which have been mentioned hitherto have all been concerned with isotropic bodies.

Cauchy in 1828[11] extended the equations to the case of crystalline substances.

This, however, he accomplished only by reverting to Navier’s plan of conceiving an elastic body as a cluster of particles which attract each other with forces depending on their distances apart.

The aelotropy he accounted for by supposing the particles to be packed more closely in some directions than in others.

The general equations thus obtained for the vibrations of an elastic solid contain twenty-one constants; six of those depend on the initial stress, so that if the body is initially without stress, only 15 constants are involved.

If, retaining the initial stress, the medium is supposed to be symmetrical with respect to three mutually orthogonal planes, the 21 constants reduce to nine, and the equations which determine the vibrations may be written in the form[12] and 2 similar equations.

The 3 constants G, H, I represent the stresses across planes parallel to the coordinate planes in the undisturbed state of the aether.[13] On the basis of these equations, Cauchy worked out a theory of light, of which an instalment relating to crystal-optics was presented to the Academy in 1830.[14] Its characteristic features will now be sketched.

By substitution in the equations last given, it is found that when the wave-front of the vibration is parallel to the plane of yz, the velocity of propagation must be (h + G)

if the vibration takes place parallel to the axis of y, and (g + G)

if it takes place parallel to the axis of z. Similarly when the wave-front is parallel to the plane of zx, the velocity must be (h + H)

if the vibration is parallel to the axis of x, and (f+ H)

if it is parallel to the axis of z; and when the wave-front is parallel to the plane xy, the velocity must be (g + I)

if the vibration is parallel to the axis of x, and (f + I)

if it is parallel to the axis of y.

Now it is known from experiment that the velocity of a ray polarized parallel to one of the planes in question is the same, whether its direction of propagation is along one or the other of the axes in that plane: so, if we assume that the vibrations which constitute light are executed parallel to the plane of polarization, we must have

This is the assumption made in the memoir of 1830: the theory based on it is generally known as Cauchy’s First Theory;[15] the equilibrium pressures G, H, I, being all equal, are taken to be zero.

If, on the other hand, we make the alternative assumption that the vibrations of the aether are executed at right angles to the plane of polarization, we must have

{\displaystyle g+G=f+H}; the theory based on this supposition is known as Cauchy’s Second Theory: it was published in 1836.[16]

In both theories, Cauchy imposes the condition that the section of two of the sheets of the wave-surface made by any one of the coordinate planes is to be formed of a circle and an ellipse, as in Fresnel’s theory; this yields the 3 conditions:

Thus in the first theory we have these together with the equations

which express the condition that the undisturbed state of the aether is unstressed; and the aethereal vibrations are executed parallel to the plane of polarization. In the second theory we have the three first equations, together with

and the plane of polarization is interpreted to be the plane at right angles to the direction of vibration of the aether.

Either of Cauchy’s theories accounts tolerably well for the phenomena of crystal-optics; but the wave-surface (or rather the two sheets of it which correspond to nearly transverse waves) is not exactly Fresnel’s. In both theories the existence of a third wave, formed of nearly longitudinal vibrations, is a formidable difficulty.

Cauchy himself anticipated that the existence of these vibrations would ultimately be demonstrated by experiment, and in one place[17] conjectured that they might be of a calorific nature.