iti the Theory of Double Refraction. 357 



or, {v^-a^) m cos X = {"d^-b^) I cos Y 



cosX _ { v''-b^)l 



cos Y ~ {v^ — a^) m ' 

 Similarly, by symmetry, 



cos 2 _ (v^ — b^) 7)1 



cSrY~ [v^-c'') n ' 

 Also, / cos X -\- m cos Y + n cos 2 = 0; therefore, 



, cos X cos Z ^ 



I ^ jfi -{- n ^ = 



cos Y cosY 



cos Z ml cos X 



cos Y n n cos Y 

 Similarly, by symmetry, 



cos Z _ I m cos Y 



cos X~ 71 71 cos X^ 



(V^-b^) p- (v'^—F-) 7f 



7-3 is h m + ;-^ — ~ — = 0, or, 



{^^—0^)7111 {^^ — c^)m 



(v'^-b'') {V'—C^) P + (w^-a2) (t;2_c2) ^2 



+ {v^'—b^) {v^—a^) 71^ = 

 / 



cos A = i — ^ 



/ r Ttrr 7t- ~\ 



Equation (D) may be put in the form 



g fm cos Z 71 cos Yl 6^ cos Y f" / cos 2"^ 

 \ cos X cos X J cos X\ cosXj 



c^ cos 2 r/ cos Y ~i 



H ^vr < XT — w ^ = 



cos X \ cos X J 



cos Z _ I m cos Y 

 cos X ~ w n cos X * 



Hence 

 cos" Y 

 cos^X 



^ cos X 



This is the equation given by Mr. Sylvester in his analy- 

 tical development of Fresnel's theory, Lond. and Edinb. Phil. 

 Mag., 1837, vol. xi. p. 464. 



In the Lond. and Edinb. Phil. Mag. for August 1838, vol. 

 xiii. p. 83. I gave a proof of the property of the conic sec- 



