370 Mr Murphy on the Inverse Method of 



(4) Inverse Method for Irrational Functions, and Functions of 



several Variables. 



12. None of the forms of <p (x) hitherto taken, have involved 

 irrational quantities, but a simple modification of the preceding 

 results, will make them include surds. By Ex. 2, 



i r -*{ hA -7)"~ 1 



if * ( *> = (^r ' then /(0 = i.a.8...(»-i) ' 



and to make this apply also to the fractional values of n, we 

 have only to put the denominator under its more general form 



/(>-• r' 



To prove this put t r+ '" = T, the limits of t are the same as 

 those of /, 



«°* ^■'■( h -'-7)""'-''-STSrX( h - '•;)""• 

 **£(^\)'"-l ("•'•')""'• 



/»- • (h.l. j)""' 

 from whence it is obvious that/(tf) = — . ,_, . 



jffv?) . 



*Ex. 9- Given £(*) = , to find/(*). 



V^t + m 



If we expand <£(.r), the form of the general term, abstracting 



a 2 " 

 from the sign, is t 2 g ^ j(a . + <w) . +4 ■ 



* Laplace, TAeor. cfo Proi. p. 97- Ed. 1820. 



To raake the limits and 1 in Laplace's formula, put e~* =/; and change the sign 

 of the integral- 



