248 Transactions of the Royal Canadian Institute 



involve the constants ar ,1 only will not be affected by the higher powers 

 of s which present themselves in the polynomial P(z,u). These 

 conditions among others will then be obtained on equating to the 

 principal residue relative to 2 = 00 in the product 



9. H{z,u)H"{z,u) 



where H"{z, u) is included under the general function H'{z, u) and has 

 the form 



10. H"{z,u)=^^^+P,{z,u) 



the function P\{z, u) being a polynomial to which we may assign a 

 limited degree in z. Under the function H"{z, u) will then be included 

 / linearly independent functions Hi{z,u), . . . ,Hi{z,u) such that the / 

 common conditions referred to above will be obtained on equating to 

 the principal residues relative to 2 = 00 in the / products 



11. H{z,u)H'^{z,u), a = l, . . . ,1, 



however general H{z, u) may be to start with. 



Again however general H{z, u) may be there will evidently be 

 included under the general function H^'^\\/z,u) functions fly^^(l/z, u) , 

 in number I, such that the principal residues relative to the value z= «» 

 in the / products 



12. H{z,u) ff(;°)(l/2,«), a = l, ...,/, 



have precisely the same linear forms in the constants a^., < as the principal 

 residues relative to the value 2= co in the corresponding products (11). 

 In the products 



13. H{z,u){h'„{z,u)-H^;'\\/z,u)), (r = l, ...,/, 



then the principal residues relative to the value 2 = « are identically 

 however general H{z u) may be. This however can only be when we 

 have identically 



14. H„{z,u)^H^^\l/z,u), tr = l /. 



The / functions H'^(z,u) are therefore conditioned not only by the 



partial basis (t)' but also by the partial basis (r)'°°l The I functions 

 in question then are built on the basis (r). On equating to the 

 principal residues relative to the value z= <» in the products (6) and (7) 

 therefore the common conditions on the coefficients of H{z, u) arise from 

 equating to the principal residues in products of the type 



15. H{z,u)H(f{z,u) 



where H'(j{z,u) is a rational function built on the basis (r). Also each 

 such independent product will furnish an independent condition since in 



