CAUCHY ON THE THEORY OF LIGHT. peri 
(a? + y? + 2°) (Pa’ + Qy + Re?) 
—[P (Q+R) 22+Q(R+P) 9°+R (P—Q) 2] pitpaey (7,) 
Now, the three circles, the three ellipses, and the surface of the 
fourth degree represented by the equations (5.), (7.), are pre- 
cisely those which Fresnel has given as indicating the course of 
the two polarized rays, as hitherto recognized in crystals having 
two optical axes; and we also know that the eccentricities of 
the ellipses are very small in these crystals. The conditions (6.) 
therefore should be sensibly verified there. Finally, it is well to 
observe, that if the eccentricities should become null, or, in other 
words, if we had 
Brae) oi) She Se ST en 
the conditions (6.) would give 
L=M=N=38,, .. . . . (9) 
and that the conditions (8.), (9.), are precisely those which ought 
to be fulfilled, in order that the elasticity of a medium should 
remain the same in every direction. 
With respect to the third polarized ray, as the intensity of its 
light is very little, it will generally be very difficult to perceive it, 
as we have already remarked. 
In summing up what has been said, we see that the condi- 
tions (6.) being supposed to be rigorously fulfilled, the sections 
made with the surface of the luminous waves by the coordinate 
“planes will coincide exactly with those given by Fresnel. As to 
the surface itself, it will be but little different from the surface of 
the fourth degree, which this celebrated philosopher has obtained, 
and consequently this last is, in the theory of light, what the ellip- 
tic motion of the planets is in the system of the world. 
The eccentricities of the ellipses, which are the sections of the 
wave surface, with the coordinate planes, being generally very 
small for crystals having one or two optical axes, it follows that 
we can determine, with a close approximation, the velocities of 
propagation of plane waves in these crystals, and the planes of 
polarization of the luminous rays, by help of the rule which I 
shall now give. 
To obtain the velocities of propagation of the plane waves 
parallel to a given plane A BC, and corresponding to the two 
polarized rays which a crystal of one or two optical axes trans- 
mits, it is sufficient to cut the ellipsoid represented by the 
equation 
