157 



(since «,., vanishes if t be an even number > 2, and rv^ ift be an odd number > 2,) we have 

 the following formulae, 



the sum being taken in each from n' = t, to »' = 2t. We have also, by (V(5)), 



J0,«') 

 '"2T,2r 



= [0] 



— 2t 



_,2n'+l n'flfF 



[0]-"'. C0]-(^"'-^^+^) 2'''. [«'-r+i]"' 



J 2t ,,2n'— 2t 



• (B(6)) 



in which, by (W(5)), 

 ^^^M^r;?^^^^^^''^'^''^'^^ ,^2.,,2n'>.2. 



i»F rf^"'. * 



(CW) 





J 2t ,,2n'— 2t 

 aa .do 



2r , „2n'-2r «'-2r/^*\2r-n'/^^\2n'-?r 



and, by the properties of factorials, 



thus the formulae (A(«>) reduce themselves to 



2t,2t * 



or, finally, effecting the summations, 



__££_ 

 "2t+1,2t+1 ~ da.db ' 



w 



2t,2t 



CO] ^[ff(^+Wj) ' ^ 

 [0] [f ] (^ + (^oi) } • ) 



{F(6)) 



