254 THE ROYAL SOCIETY OF CANADA 



(2, u) which possesses an arbitrarily assigned set of orders of coinci- 

 dence —, . . . , — ^ with the branches of the several cycles correspond- 



ing to the value z = a (or s= °°). Here a numerator o- may be or 

 any integer positive or negative. Construct a rational function in 

 the form of a product 



(13) {u-Pi) . . . {u-P'„) 



where the factors group themselves into r cycles of orders Pi, . . . , v^ , 

 corresponding to the cycles into which the branches « — Pi = 0, . . , , 

 u — P„ =0 of the equation (1) group themselves. Suppose each of 

 the series P/ to contain a finite number of terms and to coincide with 

 the corresponding series Pt out to a point beyond where this latter 

 series separates itself from all the other n — i series Pi, . . . , P^_i, 

 Pt+\, . . . , P„ . We can then evidently make the series of any cycle 

 in the product (13), each have one more coincidence with the branches 

 of the corresponding cycle of the equation (1), without affecting the 

 orders of coincidence of the product with the branches of the remain- 

 ing r — l cycles. It is then evident that we can construct a rational 

 function in the form (13) which possesses any assigned set of orders 



of coincidence — , • • • , ~" , if only these orders of coincidence 



J/l V,. 



are all sufficiently large. Whatever the assigned set of orders of 

 coincidence may be then we can construct a rational function of the 

 form (13) which possesses the set of orders of coincidence 



+i, . . . , ht 



if only the integer i is chosen sufficiently large. Therefore whatever 

 the assigned set of orders of coincidence may be we can construct a 

 rational function of the form 



{z-a)-'{u-P\) . . . {u-P'„) 



which possesses precisely the orders of coincidence in ciuestion. 



