by means of Complex Integration. 71 



and integrating over the circle r — \ a | as before, the external poles 

 are or 1 , y~ x , so that 



w w 



n , = t + i 



\«~ )\ ~x)\x~ z )\x~ w ) y \y~ )\ ~y)\y~ z )\y~ w 

 wx 2 wy 2 



(1 - x) (x-y) (1 -xz){l -xw) (1 - y) (x - y) (1 - yz) (1 - yw) 



w(x + y — xy — xyz — xyw + x % y 2 zw) 

 (1 — x)(l —y)(l — xz)(l —xw)(l —yz)(l —yw) 

 after an easy reduction. 



To test the practicability of the process I applied it to the 

 function 



(1 - ax) (1 - a 3 x) (1 - a 5 x) (l - ^Vl - ^\(l - 



a 5 



from the 12 of which, on putting y = x, we obtain the reduced 

 generating function which enumerates the concomitants of a 

 binary quintic. The result was to confirm the calculation of 

 Cayley ; the actual labour involved was, of course, considerable, 

 chiefly on account of the occurrence of poles with irrational 

 affixes. 



There is no theoretical difficulty in extending the process 

 above explained to cases where the operation fl is applied to an 

 expansion containing positive and negative powers of two or more 

 quantities a, b, etc. Thus let 



1 



/= 



(l-ax)(l-^)(l-abx 2 )(l 



then 



abj 



«v- -^ =£) 



(1 - ax) (l - -\ (1 - ax 2 ) (1 - O 



where (4) = 1 — x 4 , and 



1 

 = 



:.,';|l--j(l-atf 2 ) 



Now 



(a — 1) (a — l)(a — x)(l — ax)(l — ax 2 ) 



VOL. XIII. PT. II. 



