138 Mb GREEN, ON THE PROPAGATION OF LIGHT 



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

 obtain the fourteen relations following between the coefficients of a , 

 viz. 



o = (a^, o = oa o = ( 7 a 



= («£), = 08ft = ( 7 ,) f 

 («?) = - 2 (/3 7 ), Ifa) = - 2 (« 7 ), ( 7 = - 2 (a/3), 



(P) = W = (D = 2(« 2 ) + 0?D = a 09-) + (ft) = 2( 7 2 ) + 00l 



Hence, we may readily put the function <p 2 under the following 

 form : 



(f ) (? + «! + D 2 + (« 2 ) i« 2 - 4 ^1 + (/3 ? ) (/3 2 - * £D 



+ (7 2 ) (7 2 - 4^) + 2(/3 7 ) (07 - 2 «£) 



+ 2(« 7 )(« 7 -2/3^) + 2 (a/3) (a/3 - 2 7 0, 



or by restoring the values of f, >?, &c, and making G = {$?) L = (a 8 ) &c, 

 our function will become 



„ (du dv dw\ 2 T j (dv dw\~ dv dw \ 

 \dx dy dz) \ \dx dy I dy d%\ 



__ i (du dw\ 2 .du dw\ ^.[(du dv\ s du dv\ 

 \\dz dx) dx dz) \\dy dx) dx dy) 



pf tdu dw\ (du dv\ du (dv dw \ 1 



\\dz dx) \dy dx) dx \zd dy)\ 



(12.) 



„J/c?u dw\ (du dv \ dv (dti dw\\ 



\\dz dy) \dy dx) dy \d% dxlj 



■pUdv dw \ (du dw \ dw (du d v \\ 



\\d» dy) \dz dx) dz \dy dx))'' 



+ 



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



1 = G {ax + by + c&y + L(bz — cyf 



+ M(a% - cxf + N(ay - bxf + 2P(cx - az)(ay - bx) 



(13.) 

 + 2 Q (bz — cy) (ay — bx) + 2 B (bz — cy) (ex - az). 



