Roberts — 0>i the Reduction of an Integral. 95 



^ ' • /(s) /(z) 



and introducing into tliis the values of and ; — ^^ from the 



z-n {z-n)~ 



•equations (1) and (2) we arrive at the formula 



,(3) 1 455._4^f^_^i _ /w /(^) 



"We have denoted the integral p=: bv Z ; 



•we have consequently -^ = -= • 



dn ] ,^{z-nyjf(,) 



introducing these values into equations (3) after integrating with 

 regard to z, we obtain 



<^) /;;S =* r^' A") f -*/(»)-. 



where, for the sake of brevity, we have written 



dz dn 



Now R (z, n) is of the 2m -2'"^ degree in z; consequently integrals 



of the type 



^ dz 



are, by this formula of reduction, shown to depend on the 2m - 1 I, 

 and the integral Z , ?^ 



Differentiating this formula of reduction with regard to n, we 

 readily perceive that the integral 



dz 



depends on 



(s - nyjf{z) 

 dz 



{z - nrJM 



