[ 9'^ ] 



IX. 



ON THE EEDTJCTIOjS" OP THE INTEGEAL [^il^ 

 TO A NUMBER OF OTHEE INTEGEALS OF THE FOEM 



f ^-P= WHEN <i(s) AND xl/U) AEE EATIONAL AND 



INTEGEAL FUNCTIONS OF % AND /(z), A POLYNOMIAL 

 OF THE DEGEEE 2m. By EEV. W. E. EOBEETS, F.T.C.D. 



[communicated by EEV. JOHN H. BEE.XAKD, D.D.] 



[Eead January 22, 1900]. 



1. In the discussion which, follows we shall assume that the roots 

 of /(z) = 0, which we shall denote by ai, a^ . ■ . a,,,,, are real, and also 

 that the limiting values of s in the integrals treated of are also real. 

 "We shall write, then, 



/(z) - z^'» + P^-' + P^^'"'-' + . . . + P2„. - (z - aO(z -a,)...{z- a,„,). 



"We shall now let 



r being an integer, or, simply /„ and Z, when we do not wish to put 

 the elements in evidence. 



2. We now proceed to classify the various elementary integrals 



C di(z)dz 

 upon which the general integral — , discussed in this paper, 



can be made to depend. 



Let us, in the first instance, suppose that the degree of (f> (z) exceeds 

 that of ij/ (z) ; in this case, it is well known that jjj: consists of a 

 number of terms of the type c,z', c^ being a constant, and of a number 



