7 
axis, these two semi-diameters, or their squares Q, R, always 
differ very little one from the other, and that in consequence 
the generating ellipse of the ellipsoid exhibits a very small eccen- 
tricity. Hence it also results that the condition (2.) sensibly re- 
duces itself to the following : 
NSS; 
that is to say, to a condition which is fulfilled, whenever the 
elasticity of a medium continues the same in every direction 
around any point. We may add, that the intensity of the light 
determined by calculation for each of the two polarized rays 
which we are here considering, is precisely that which obser- 
vation furnishes. As to the third polarized ray, calculation 
shows that it is very difficult to perceive it, inasmuch as the 
intensity of the light continues always very small in it when it is 
not rigorously nothing. We will hereafter seek the means of 
proving its existence. 
Let us now suppose that, in the ethereal fluid, the elasticity 
ceases to be the same in all directions around an axis parallel to 
the axis of z. If we cut the surface of the luminous waves by 
the coordinate planes, the sections made with two sheets of this 
surface may be reduced to the three circles, and to the three 
ellipses represented by the equations 
276 CAUCHY ON THE THEORY OF LIGHT. 
2 22 “ 2 =i 2 
Rta=®% ss dpediai: 
2 x : 2 = a 
pra 'ts Q s2*s “Vianeam (5.) 
x 23 r at 2 
atpa® R= 
and, in order that this reduction may take place, it will suffice 
that the coefficients G, H, I, being evanescent, the three condi- 
tions 
(M—P) (N—P)=4P*, (N-Q) (L—Q)=4 pres 
(L—R) (M—R)=4R, » (6.) 
all three similar to the condition (2.), be verified. Furthermore, 
if the eccentricities of the three ellipses are small enough for 
their squares to be neglected, the conditions (6.) will lead to the 
following : 
(M—P) (N— Q) (L—R)=(N—P) (L—Q) (M—R) =8 PQR, 
and the equation of the wave-surface may be reduced to 
