Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/163

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.^[1] 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)^1/2 if the vibration takes place parallel to the axis of y, and (g + G)^1/2 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)^1/2 if the vibration is parallel to the axis of x, and (f+ H)^1/2: 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)^1/2 if the vibration is parallel to the axis of x, and (f + I)^1/2 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

$f+H=f+I$, $g+I=g+G$, $h+H=h+G$;

or, $G=H=I$. This is the assumption made in the memoir of 1830: the theory based on it is generally known as Cauchy's First Theory;^[2] 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 $h+H=4+1$, $f+I=h+G$, $g+G=f+H$;

1. Mém. de l'Acad., x, p. 293.
   In the previous year (Mém. de l'Acad., ix, p. 114) Cauchy had stated that the equations of elasticity lead in the case of uniaxal crystals to a wave-surface of which two sheets are a sphere and spheroid as in Huygens' theory.
2. The equations and results of Cauchy's First Theory of crystal-optics were independently obtained shortly afterwards by Franz Ernst Neumann (b. 1798, d. 1895): cf. Ann. d. Phys. xxv (1832), p. 418, reprinted as No. 76 of Ostwald's Klassiker der exakten Wissenschaften, with notes by A. Wangerin.