146 Mr. Murphy on the 



SECTION V. 



Let F(x) be any function of x, containing only integer 

 powers of x, positive, or negative ; as 



F(x) = A + Bx + Cx" + &c. 



+ - + — + &c. 



X X* 



Put e" for x, multiply then by e e and integrate with respect to 9, 

 we thus obtain 



/> ( e «). 6 « = const. +^6 9 +^J- + -|- + &c. 



+ b.9-ce~ 6 - &c. 



All the terras in this result, except b.9 and the const., are cir- 

 culating or periodical terms; i.e. if we give 9 the series of ima- 

 ginary values comprised in the formula 9 + a^/ — 1, then for any 

 two values of a, which differ by (2ir) a whole circumference, 

 these terms have precisely the same value, but the term b.9 be- 

 comes 2wb^/ -1 between the above limits: thus it appears that 

 the coefficient of the first negative power of x in 



S57T +J — 1 



the integral being taken through one entire circulation, i.e. from 



9 to 9 + 2tt v /^1. 



Restore now for e 9 its value x; then f t F[e i ).e i becomes f a .F(x) 

 which by actual integration is 



