246 Transactions of the Royal Canadian Institute 



We shall designate by H'{z,u) and H'^°°\1/z, u) the general rationa 



functions of {z,u) conditioned by the partial bases (r)' and (t)^°°^ 



respectively. The writer has shewn* that the necessary and sufficient 



conditions in order that a rational function H{%, u) may have the orders 



of coincidence F i*\ . . . . ,7"/*^ for the value z^a^ are obtained on 



k 



equating to the principal residue relative to z = ak in the product 



4. H (z, u) Hk (z, w) 



where we employ the notation Hk (s, u) to designate the general rational 

 function conditioned by the orders of coincidence t[\- ■ ■ ■, t for the 

 specific value z = Ok . It is also readily shewn that we impose on 

 H{z, u) the orders of coincidence Fi, . , . . ,7>^ relative to the specific value 

 z=ak when we equate to the principal residue relative to the 

 value z = ak in the product (4?), after we have imposed on the factor 

 Hk (2, u) any assigned set of conditions relative to finite values of the 

 variable s other than the specific value here in question. In particular 

 we see that we impose the orders of coincidence •'^1 »•■••> t ori the 

 function H{z, u) when we equate to the principal residue relative to 

 z = ak in the product 



5. H (z, u) Hk {3, u) 



where H'J^z, u) designates the most general function included under 

 H' (3, u) such that the principal residues in the product relative to all 

 finite jyalues of the variable z other than z = ak vanish identically. 

 Here H{z, u) is supposed to be a form as general as we will but chosen 

 in advance. Thereafter i?^(?. u) is chosen accordingly and is included 

 under H'{z,ii). In the product (5) then there can only be question of 

 principal residues relative to the specific value z = ak and to the value 

 z = 00 . If one of these residues is then the other must be also. On 

 equating to the principal residue relative to the value z= 00 in the 

 product (5) therefore we at the same time equate to the_principal 

 residue relative to the value z = ak and therewith impose on H{z,u) the 

 orders of coincidence ti, . . . , Tr;^ for this value of the variable. 

 Consequently on equating to the principal residue relative to the 

 value z= c» in the product 



6. H(z, u) H'{z, u) 



we impose on H(z, u) the orders of coincidence ri, . . . , Tr;^, relative to 

 the value z = ak . The principal residue relative to z=ak in the product 

 (6) will then be 0. This holds for every finite value z = Uk . On 



*0n the Foundations of the Theory of Algebraic Functions of one Variable. Trans. 

 Roy. Soc, London, Series A, Vol. 212, p. 347. 



