362 Mr MURPHY'S THIRD MEMOIR ON THE 



Examples : 



/» - -J_ V _ ^-3 V J. g-4-5 ^ . 



p_j^ 2«+3 „ (2w+4)(2«+5) (2m+5)(2« + 6)(2w+7) .^ , 



-rn—r„ J . ;'„+,+ I ^ . f'n+a J . f'n+g&C. 



the latter series would also result by reverting the series for V„, in 

 Art. 24. 



30. To find a function U„ which shall he reciprocal to (h.l.ty. 



Following the steps indicated in Art. 23, we must first form a self- 

 reciprocal function of which the general term is a constant multiplied 

 by (h. 1. ty ; this has been already effected in Sect, v, namely, 



and then the form of the required function will be 



[/„ = t; + a 2;+ , + 6 1;^^ + c 7;+3 + &c. 



Multiply by (h. 1. ty, supposing m>n, and observing that 

 j;r„(h.l.0" = 1.2.3...>^.(-ir. "-^'^-;)^:-f-("-'^-^^) bySect.v, 



and ir7„(h. 1. /)•" = 

 by the nature of reciprocal functions, we get the general identity 



m{m — \){m — Q)...{m — n + l) , m — n , {m-n){m -n—\) 



® r.2.3...« •^^""•^TT+*- (« + i)(« + 2) — *'*'-^' 



but on the same supposition that m is greater than w, we also have 

 = (l-l)'"-" = l-(m-w)+^^ 4p— ^-&c.; 



