IN CRYSTALLIZED MEDIA. 309 



Substituting in these the values of A, B, &c., before given, 

 we shall obtain the fourteen relations following between the 

 coefficients of c/> 2 , viz. 



0=M, =(), 0=( 7 ?), = 



(*?)=- 2 (07), (0*)=-2(7), ( 7 ) = - 2 (a/3), 

 (f) = W = (D = 2 (a 2 ) + faf) = 2 (/3<) + (#) = 2 (7 s ) + 



Hence, we may readily put the function < 2 under the follow- 

 ing form, 



2 + (a 1 ) (a 2 - 4*) + 08") (/S 2 - 4#) + ( 7 2 ) (7 2 - 

 + 2 (a 7 ) (a 7 - 20*) + 2 (a/3) (a0 - 



or by restoring the values of f, 77, &c., and making G = 

 L = (a 2 ), &c., our function will become 



r tdu, dv dw\* 



(JT lj -- T -r + -7- + 



\dx dy dzj 



.dvdw du dw\* dudw 



p (fdu dw\ tdu dv\ du fdv dw 

 + + ~ 2 ~ + 



+ 



dy dy + dx dy dz dx 



....... (12), 



dy) \dz dxj dz \dy dx)} 



and hence we get for the equation of the corresponding ellipsoid, 

 1 = G (ax + ly + czf + L(bz- cy}* +M(az- cxf + N (ay - lx) z 



But if in equation (8) and corresponding function (^4), we 

 suppose .4=0, B 0, and (7 = 0, and then refer the equation to 

 axes taken arbitrarily in space, we shall thus introduce three 



