378 Mr Murphy on the Inverse Method of 



. 1 1 R 8«aff 1^3 2 4 («/3f 



where if.-U+A)*" 



f_l 1 (<2.n+\).tfh 1 (2» + 1).(2m + 3).2V< 9 1 „ I 



x i(l-A) s " +1 "w + l a(l-*) fc+8 *» + a 2.4.(1 -h)"+ a 'n + 3 J 



= f w (1 7^ / 'C';^ n + i from '=° to '- 1 - 



Jt {{\ — hf +2 2 «q n + i 



20. To find a formula, which shall represent -^+n when «>/3, 



171 



and ^n, when /3>a. 



Suppose that H„ is the coefficient of h m in 2?„, in the last 

 Article, it follows that 



1 1 2*a/3 1.3 „ 8'(«/3)« , 



is the coefficient of hr in the least of 



-~i or ^r, according as a > or < /3. 



" . , i. e. it represents 



l/3-A« 



Integrating by parts, and observing that 2? = the least of 



r 1 

 j- f 1 l+A aa.d+A)' 



1 = 1 5 we get ■"•"-'iJKZT'ir (2»-l).(2«-3).(2 3 /V) 



-A 



a».(2»-a) (1 + *)' & l 



"*'(aB-i).(a»-s).(a»-5)" 2 5 /* 3 Tw ""j 



2.4.6.. ,2w (1+k)* 

 + 1.3.5. ..(2»-l)" 2" n A" 



