418 Mr green, ON THE DETERMINATION OF THE 

 ^ ' [ a^b^ aa^\^^^ a^ c^V af (n) J 



(1 - 2P^|-422— i^^^^— _22-L ^ 1 



P g rf'0 P (/'^ 



2(l-2f)2JL.^ +22i:x2^^^ 



-2 (1-2^2^x2— ^ -2 2-x2^x2-^ 



all the finite integrals being taken from ?• = 1 to r = s + 1, and from 

 / = 1 to r' = s + 1. 



In order to obtain the required value 



v'v^ - w'<p, 



it is clear that we shall only have to add the first of the five preceding 

 quantities to the sum of the four following ones multiplied by A', and 

 to render this more easy, we have appended to each of the terms in the 

 preceding quantities a number inclosed in a small parenthesis. 



Now since the accents may be permuted at will, and we have like- 

 wise or = b^ + U, it is easy to see that the terms marked (1), (6) and 



(12) mutually destroy each other. In like manner, (2), (3), (7) and 

 (18) mutually destroy each other; the same may evidently be said of 



(13) and (16), of (15) and (17), of (9) and (19), and of (8) and (14). 

 Moreover the four quantities (4), (5), (10) and (11) will do so likewise, 

 and consequently, we have 



V'V0 - VV> = 0. 

 Hence the truth of the equation (31) is manifest. 



