the theory based on this supposition is known as Cauchy's Second Theory: it was published in 1836.[1]
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 three 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[2] conjectured that they might be of a calorific nature. A further objection to Cauchy's theories is that the relations between the constants do not appear to admit of any simple physical interpretation, being evidently assumed for the sole purpose of forcing the formulae into some degree of conformity with the results of experiment. And further difficulties will appear when we proceed subsequently to compare the properties which are assigned to the aether in crystal-optics with those which must be postulated in order to account for reflexion and refraction.
