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 #, y, s, 

 we have, by equation (2), 



+ b z PS Y + c*Z3Z ; (3) 



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

 the characteristics d and S, 



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*Z8n - b*Y$Z) + JJ dxdz (a z 



(4) 



j -- u ~J~ *** "^ I a ~j c j 



dy dz J \ dz dx) 



:r - - 

 d# dy 



But as the variations 85, 817, S? 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 



