126 ON THE LAWS OF 



the integrals being taken from r=r to r' = l, from 0' = to 

 & = 7r, and from v?' = Q to OT' = 2?r. This will be evident by 

 recollecting that the triple integral by which the value of V" is 

 expressed, is the same as the one before given for V, except that 

 the integration, relative to r, instead of extending from r' = to 

 r = 1, ought only to extend from r r to r = I. 



But the general term in the function f(r' 2 ) being represented 

 by A t r' 2t , the part of T 8 dependent on this term will evidently be 



A t r jdV a<v' sin 6'. QJr'-*- n dr' (1 - r'^ (2) ; 



the limits of the integrals being the same as before, 



We thus see that the value of T 8 and consequently of V" 

 would immediately be obtained, provided we had the value of 

 the general integral 



JV'*-'(t-T. 

 which being expanded and integrated becomes 



i 



but since the first line of this expression is the well-known 

 expansion of 



- ) or 

 <1> 



r W 



nj \nj 



nr 



when n = 2.p = b+l and q = 2 (/3 + 1) we have ultimately, 



By means of the result here obtained, we shall readily find 

 the value of the expression (2), which will evidently contain one 

 term multiplied by r 8 and an infinite number of others, in all of 



