452 Mr. R. Hargreaves on 



centre of an ellipsoid u a — l ; the comparison of the direct 

 integral and the solution gives 



kp 



CCidx dy dz kp f x d\ CCCdx dy dz f °° d\ f . ao . 



and the volume-integral embraces all values for which u a ^ 1. 

 For various points on a line in the direction (linn), the ratio 



s/u-p : r is constant, and = \/u v (I, m, n). We may therefore 

 integrate with regard to r for a cone of fine angle dco and 

 axis in the direction (Imn), up to the surface where 



~ 2 =u a (l, m, n). Thus 



r 



i 



1(0 



fdx dy dz__ _ C C r 2 drdco _ C r,V<a _ C t 



s/u P (x,y,z) Jjr \/uv(l,m,n) j2 y/u F (l,m,n) J$Ua 



and so 



s/ 



Uv 



M « V«P Jo 



v/J 



is the equivalent of (123), when I, wi, n are variables in w ffl 



and //p, which need not be written as the angular integral is 

 always shown by dco. The value of J being 



1 + \1 (pa + 2p'a') + X 2 S (PA + 2 P'A') + X 3 A p A a , 



it is not altered if a . . . and p . . . are interchanged. Thus 

 we have the double form 





In (123) there is also an alternative form 



\- — tt subject to u.p ^> 1. 



In the original problem of solving a differential equation, 

 a . . . and p . . . appeared as primary constants, i. e., a . . . 

 defined the ellipsoid in reference to which a solution was 

 sought, and p . . . defined the differential equation. In (124) 

 a and P occur as primary constants in the first case, and 

 p and A in the second or reciprocal case. To meet this we 

 may return to the original determinant A (X) = A 2 J(X), and 

 at the same time use /a = XA a for parameter ; i. e., 



\ (125) 

 where A (fi) is the determinant A H- pyu, ... J 



