Direct Derivation of the Complementary Theorem 249 



any such product ^he principal residue relative to 2 = 00 is not 

 identically when H{z, u) is taken sufficiently general. On desig- 

 nating by Nj and Nf respectively the numbers of the arbitrary 

 constants involved in the general rational functions built on the bases 

 (r) and (7) we have 1 = Nt and the number of the conditions common 

 to those imposed on the function H(!z, u) by the partial bases (t)' and 

 (7)(°°) respectively is precisely N^ . 



Suppose {t) to be a basis on a lower level than either of the bases (t) 

 or (t). We start out with a sufficiently general rational function 

 H{z,u). In particular we assume this function to include the general 

 rational function built on the basis (/). The numbers of the conditions 

 imposed on the function H{z,u) by the partial bases {t)' and (/)^°°^ 

 respectively we shall indicate by Mt and M)^'. The numbers of the 

 conditions imposed on H(z,u) by the partial bases (V)' and (7)^*"^ 



will then be M[+ 2' 2* (f/*> - /f ) ) j/f > and ilf^ °° ) + s'" (yl °° ) _/( °° ))^^( « 



k s = l s=l 



respectively. From what precedes then the number of the 

 independent conditions imposed on H{z, u) by these partial bases 

 simultaneously, that is by the basis (7), will be 



16. M7 = M[-\-Ml'^^-h S S* (7f -ft vl''^-Nr . 



k s=l _ 



Similarly the number of the independent conditions imposed on H{z, u) 

 by the basis (t) will be 



17. Mr ^M't^-Ml"^^ + S 2* (r?'-/f )''?^-iVf-. 



k s=l 



Evidently Mr -{-Nt =MT-\- Nt since these expressions both represent 

 the number of the arbitrary constants involved in the original function 

 H{z,u). We therefore have Mr — Mt = Nf— Nt and by combining 

 (16) and (17) we obtain the complementary theorem in the form 



18. Mr -h ^i'^rfV^^^MT-h 2 2* T^vf^- 



ks=l k s=\ 



