420 On the Wave-surface in the Theory of Double Refraction. 

 d a e v d*£ v d 3 >j „ d 2 ? 



Tt* = cos x d> + cos Y dF + cos z if? 



~f- = «fa 2 cos 2 X + 6 2 cos 3 Y + c 2 cos 2 s}-0 



+ /a 2 cos 8 X + b 3 cos 3 Y + C 2 cos 2 *}- ^ 



If *> 2 = « 9 cos 3 X + 6 3 cos 3 Y + c 2 cos 9 % 



ti 2 = a 2 cos 2 X + b 2 cos 2 Y + C 2 cos 2 2 



d ** d r* + U d r* * 



This equation is readily integrable if w and il are constant, 

 that is, if the direction of the displacement be invariable ; and 

 p may be expressed by a |eries of terms similar to 



(a sin kr-\-b cos Jcr) sin n t -f («' sin Jcr + U cos £ r) cos w £, 

 and represents a wave of light moving in the direction of r 



with the velocity -j- 9 



k ~ V \ 2.3.4 t; 2 J ' 



See Mr.Tovey's excellent paper, L.&E. Phil. Mag. January 



1836. 



Fresnel does not implicitly take into consideration the term 



fo 2 d*e 



4 % which according to M. Cauchy is necessary to explain 



dispersion. We have then 

 d 3 e „ d 2 



9 2 a 9 



- — V 2 *r4r V = 



d* 2 dr 2 k ' 



Fresnel supposes that the quantity v 2 will be constant when 

 the force revolved in a direction perpendicular to the direc- 

 tion of displacement A g is also perpendicular to a given plane, 

 and he shows that this will be the case precisely when the 

 quantity v 2 = a 2 cos 3 X+b* cos 2 Y+c 2 cos 3 Z is a maximum 

 or minimum. If the given plane is parallel to the plane 



nx+ny+lz — 0. 



Fresnel finds by geometry and without further assumptions 

 the equation 



(a 2 -tt 3 ) (c 2 -uO w 2 + {b*-v 2 ) {c*-v 2 ) m 1 



+ (« 2 -u 2 ) (b 2 -v 2 )l* = 0. 



The calculation of Fresnel's equation to the wave-surface 



is also now a purely geometrical problem which is easily 



effected by the elegant artifices suggested by Mr. Smith in 



