Crystalline Reflexion and Refraction. 157 



SECT. IV. PROPAGATION OF LIGHT IN A CRYSTALLIZED ME- 

 DIUM. LAWS OF FRESNEL. ALTERATION REQUIRED TO BE 

 MADE IN THEM. WAVE-SURFACE, INDEX-SURFACE, AND 

 THEIR PROPERTIES. 



The principal axes of the crystal being the axes of x, y, z, 

 we have, by equation (2), 



- SV = a?X$ X +PYSY+ c z Z3Z ; (3) 



or, by taking the variations from formulae (c), and interchanging 

 the characteristics d and 8, 



- - - 



dy J \dx dz J \dy dx ) 



and if we substitute this value in equation (1), and then inte- 

 grate by parts the right-hand member, in order to get rid of the 

 differential coefficients of the variations, we shall obtain 



- 



= JJ dydz (c*ZSn - b*YSZ) + JJ dxdz (a* 



+ JJ dxdy (b 2 YSZ - a z XSr,) (4) 



mj ^^ tf * dz d * 

 dxdydz c 2 -V - 



(\ dy dz 



\ dz dx 



dx 



But as the variations 8$, Sij, 8^ are arbitrary and independent, 

 this equation cannot hold unless the double integrals, which re- 

 late to the limits of the system, reduce themselves to zero, leav- 

 ing the equality to subsist, independently of the variations, 

 between the triple integrals alone. Equating, therefore, the 



