974 CAUCHY ON THE THEORY OF LIGHT. 
seen the simple ray disappear when the initial vibrations of the 
molecules of the ather were supposed perpendicular to the di- 
rections of the rays, and then, properly speaking, there was no 
more polarization. Now, the reduction of all the rays to a sin- 
gle one, and the absence of all polarization in the media in which 
the light remains the same in every direction, being verified by 
experiment, we have drawn this definitive conclusion from our 
analysis, that, in ordinary light, the vibrations are transversal, 
that is to say, perpendicular to the directions of the rays; and 
thus the hypothesis which Fresnel proposed, notwithstanding 
the arguments and calculations of an illustrious adversary, has 
become a reality. 
We shall now briefly apply the theory which we have just 
recapitulated to the propagation of light in the crystals having 
one, or two, optical axes. In order to effect this, it will not be 
necessary to employ the general equations which we gave in the 
Thirty-first Part of the ‘ Exercices,’ to represent the motion of 
a system of molecules acted upon by mutual attracting or repel- 
ling forces, and these equations may be reduced to the formule 
(68) of page 208 of the Third Volume; that is to say, to the for- 
mule which express the motion of a system which offers three 
axes of elasticity perpendicular one to another. We may more- 
over suppose that no force arising within itself is applied to the 
system, and then the formulz in question will only include the 
time ¢, the coordinates 2, y, z of any molecule m, its displace- 
ments £, 7, €, measured parallel to the coordinate axes, and nine 
coefficients G, H, I, L, M, N, P, Q, R, the three first of which 
are proportional to the pressures sustained in the natural state 
of the ethereal fluid, by three planes respectively perpendicular 
to these same axes. The coefficients here in question being 
considered as constant, we shall easily construct the ellipsoid 
whose three axes are reciprocally proportional to the three ve- 
locities of propagation of the plane waves parallel to a given 
plane, and drawn parallel to the right lines, according to which 
are measured the proper velocities of the azthereal molecules in 
these plane waves. We may also determine, Ist, the directions 
of the three polarized rays produced by the subdivision of a 
luminous ray in which the vibrations of the molecules would 
have any directions whatever; 2nd, the velocity of the light in 
each of these three rays; 3rd, the different values which this 
velocity would take in the polarized rays produced by the sub- 
