Direct Derivation of the Complementary Theorem 247 



equating to the principal residue relative to the value 2=00 in the 

 product (G) therefore we impose on H{z, u) the orders of coincidence 

 furnished by the partial basis {/r)' and equate to the principal residues 

 in the product relative to finite values of z. The vanishing of the 

 principal residue relative to the value z= 00 in the product (6) then 

 furnishes the necessary and sufficient conditions in order that the 

 function H{z, u) may have the orders of coincidence required by the 

 partial basis (t)'. 



The necessary and sufficient conditions in order that a rational 

 function H{z,u) may have the orders of coincidence ~t°^ , . . . ,'t. for 



the value s= 00 are obtained on equating to the principal residue 

 relative to the value 2= 00 in the product 



7. H{z,u)H^'^\\/z,u). 



The necessary and sufficient conditions in order that a rational function 

 H{z, u) may be built on the basis ( t) are obtained on equating to the 

 principal residues relative to 2= 00 in the two products (6) and (7). 

 The question arises as to how far the two sets of conditions on the 

 coefficients of H{z, u) so obtained overlap? To determine this we write 

 H{z, u) in the form 



8. ^(2,w)=2-'-i((l/s,w)) + "s S ar,tz' u' -{- Ji{z-Ok)^kPi{z,u) 



t = r= - i k 



where Pi{z,u) is a polynomial in (2, w) and where j + 1=2 4 . The 



k 

 exponents 4 we assume to be chosen sufficiently great to ensure that 



the product IT (2 — c^ )** Pi (2, u) satisfies the conditions imposed by the 



k 

 partial basis (t)' and i we take sufficiently great to ensure that the 



orders of coincidence of the expression 2~*"'^((l/2, u)) for the value 



2 = 00 do not fall short of those furnished by the partial basis (7)^°°^ 



Now on substituting in the products (6) and (7) the expression for 



H{z, u) given in (8) we see that the principal residue relative to 2 = 00 in 



the former product does not involve coefficients in the expression 



n ( 2— QkYk Pi(z,u) and that the principal residue relative to 2= 00 in 



k 



the latter product does not involve coefficients in the expression 



2~*~K(l/2, w)). The conditions common to the two sets of conditions 



obtained on equating to the principal residues relative to 2 = 00 in the 



products (6) and (7) then evidently involve only the coefficients a^, < in 



the expression for H(z,u) given in (8). Supposing / to be the number 



of these common conditions and writing H'(z, u) in the form for a 



rational function given in (2) we see on equating to the principal 



residue relative to 2 = co in the product (6) that those conditions which 



