159 



n 



by partial differential and factorial transformations, similar to those which we have already em- 

 ployed in finding the sums of the expressions (BM). Thus, we may eliminate the variable 

 n' from the partial differential index, by putting, in virtue of (WW) 



ro]-<2r+l)ro-j- «' \ -^ da) _ r2«'_2Tl2»'-2'X 



.[O]-'.[O]-(3-')[O]-''.[O]-('"-i-»0f i.-ll) (~] V da J (K(60 



\ da-'/ \da.db) da^.dh' 



the integers s, s', being new variables connected by the relation 



s + s'=:2n'— 2t, (LCS)) 



which gives, by the properties of factorials, 



l2n'—2ry'^-'>^ [0]-^. = [2»'— 2t]' = 2'ln' _r]' + 2'-2[,]»[„'_x]*-i. (Mt«)) 



observing that by (KC«)), s is included between the limits and 3 ; and, by the same pro 

 perties, 



[0] 



t' being an arbitrary integer; so that if we put 





(NW) 



\c?aV KdaMj 



the expression (IW) resolves tself into the two following parts : 



I 



(OW) 



(Pm) 



St"') w 



da^^-' db' 



d= f^ + F.-^) 



in which £("0, 2W, denote summations with reference to the two independent variables n' and s, 

 and which can be calculated separately, by making in the first t" = 1, and in the second ■/ = : 

 for this gives, by the binomial theorem, 



TOL. XV. A A 



