72 Mr Mathews, Redaction of Generating Functions, etc. 



and, calculating the residues for the poles 1 and x, we obtain 



1 x 1 + x 2 



£l a cf) = z,R ( 



(1)»(2) (1)(2)(3) (1)(2)(3)' 



and hence 



1 



^a,bf=JT\^a4> = 



(4) ^ (1)(2) 2 (3) 



(1 - x) (1 - a; 2 ) 2 (1 - ^) * 



There is one very curious point about this theory which 

 deserves to be noticed. In the actual applications the letters 

 used are mere umbrse, and the question of their numerical values 

 does not come in ; but when, as here, the calculus of residues is 

 employed, it is necessary to assume relations of numerical in- 

 equality for the purpose of deciding which poles are to be con- 

 sidered as being inside, and which outside, the contour of 

 integration. 



It cannot, I fear, be claimed for this method of residues that 

 it supersedes others, such as those employed by MacMahon in his 

 papers on partitions : but it is, at any rate, a useful auxiliary, and 

 the fact that it is theoretically complete entitles it to some 

 consideration. 



