322 Mr MURPHY'S THIRD MEMOIR ON THE 



7. For the purpose of convenience both in evaluating and using 

 reciprocal functions, the knowledge of the functions which they generate 

 is very useful. The generating function, for example, being the quan- 

 tity denoted by q^, Art. (6), the process for finding in this case the 

 function generated, will sufficiently exhibit the general principle, and 

 therefore it is now proposed tb sum the series q^ + q^h + q^k' + q^h^, &c. 



Substituting for q„ its value given in the preceding article, and 

 representing the required sum by S we have 



o /. 'V J. ditty-'" le d'itt'f-'" M dHtfy-'" , , 



But if we form the equation, u = t + ku (1 — u), and suppose y'(M) to 

 be the derived function from J'{u), we have generally 



^^ r(«\-f'(A^h^it^^)-^A. *' d^{f'it).(ttj} 



,_f^_ d?\f{t).{tty\ 



+ 17273 • df *'''■ 



which is obtained by differentiating the value of /(«) given by La- 

 grange's Theorem. 



The preceding series coincide by supposing 



f(f) = {ttf)-"' = t-" il-t)"", 



and therefore /'(«) = «"" (l-w)"" = j^-L 



by the assumed equation. 



(u-t)-'" du 

 Hence 5- = -^^ . ^^ . 



Now the actual solution of the assumed quadratic equation gives 



u = 



2h 



, where R= {l-2h{l-2t) + h'}K 



, B-l-\-h{l-2t) . du 1 

 whence u-t= ^ , and -^ = ^i 



