364 Mr Murphy on the Inverse Method of 



S^e j£L-- {l-*h.L (0 + ^.(h.l..^-&c.| 



x {x--n.mx— 1 + "(*'l 1 ) .ntx— '-&c.|, 



X •2a 



the coefficient of - = ■ \ Q * ,. .{l+mhJ.(Q+ ™' ( . h - U >' +& c .j 

 a; 1.2. 3. ..(«— 1) ' ' 1.2 



.(»-l) 



K)' 



1.2.3....(»-1)' 



r- 



and therefore in this case f(t) = — — - / ' ,, 



17 ' 1.2.3 ....(re — 1) 



r-.(h.l.i)- 



Ex.3. Given f t f(t). * x =h. 1. (l + — ^— )> »» being essentially posi- 

 tive, and « either positive or between and — m. 



In this case we have 



^••^L(l + ^)-r.{l-(» + «)LL(<)+^H^;(h.L/)f-kc.J 



f_a 1 a* o 1 



U+»« 2*(x+»j) 8 <sc- r 



and if we select the coefficient of - from the products of the 



Ob 



corresponding terms in both series (which are the only products 

 that contain -) we get, the coefficient of - in 



h.\.(ty 



t°— i 



and therefore f( t ) = tm ~ 1 -\TT17\' 



