FRESNEL ON DOUBLE REFRACTION. 301 



The equation (A.) gives [v^) as a function of (m) and (n). If 

 we make (m) and (n) vary successively by a very small quantity, 

 we shall have two new tangent planes very near the former, and 

 the common intersection of these three planes will belong to the 

 surface of the wave. We must then, in the first place, differen- 

 tiate equations (A.) and (B.) with regard to (m), supposing {n) 

 constant, which gives 



{z — m X - ny) x + u^ m + (1 + m^ + n^) ^ = 0. . . (B'.) 



dm ^^ 



- [b'-u^) (c2-o2) m = (A'.) 



Differentiating afterwards with regard to {n), without making 

 (m) vary, we find in the same way, 



{2-i7ix-ny)ij + v'^n + {l+ni^ + n^)-^ = 0. . . . (Bi-) 



"^ [ (1+^2) («-2_„2) + (1 +„,2) (5-2_„2) + {m^ + n^) (c2_u2) ] 



- («2_u2) (c2_u2),^ = (A,.) 



If we now eliminate V— between equations (A'.) and (B'.), and 

 dm 



- — between equations (Aj.) and (B,.), two new equations will be 

 dn 



obtained, containing only the variable quantities (u), {m) and (ra), 

 besides the rectangular coordinates s,y,z; and joining them to 

 equations (A.) and (B.), we shall have four equations between 

 which we may eliminate u, m and n. The relation obtained by 

 this ehmination between the coordinates x, y and z will be the 

 general equation of the waves, and will belong at the same time 

 to the surface of the ordinary wave and to that of the extraor- 

 dinary wave. 



Another method of calculating the surface of the waves. 



This direct method seems necessarily to lead into calculations 

 of harassing length, in consequence of the number of quantities 

 to be eUminated and the degree of the equations. We may, it 

 is true, eliminate {v^) between equations (A.) and (B.) before dif- 

 ferentiating them, which gives an equation of the fourth degree 

 in (m) and (n). 



A more simple equation, and of the third degree only, is 



