330 Mb MURPHYs THIRD MEMOIR ON THE 



5.913...(4« + 1) 



q. 



4.8.12 4« 



, (,0J {t-- (4>^ + l)-4^,.-.,^ (4. + l).4^».^(4«-3)(4»-4) ^,„.,^, ^^^^ 



(2.) When m= — 1. 



In this case <?„ = 1.2.3...,,^^, 



and <^ = j;(«')-'=h.l. (I). 



Hence, ^ = e*, ^ = 1 + e* ; 



therefore 



_(« + l)_ ^^ ^+1 nin-X) {n^\){n) ^_^ , 



9» (1 + £</>)»+'= »^ 1- 2 -^ ^ 1.2 ■ 2.3 "^ '^'^■^' 



where the limits of ^ are — « and + w . 



13. To express the functions Q^ and qn z« terms of t alone. 

 By Art. 6., we have 



^'~ 1.2.3...ndf 



= (w-?»)(w-OT-l)...(l-OT) _ „_ 71 n-m 



1 .2.3...W v ; • J j^ . ^ _^ 



«j^_l) (/^-m)(«->»-l) , 



^ 1.2 • (l-»»)(2-»/) ^ ^>&c.}. 



