600 TRANSACTIONS OF SECTION A. 



where | = «-', t) = s""' v, and s'" is the highest power of ;: which appears in the 

 equation. If the function is to become infinite to orders not higher than 



respectively for branches of the cjcles corresponding to finite values s = a,,, and 

 if for 5 = o: it is to have orders of coincidence which do not fall short of 



' oo 



respectively, then will the functions ^ contain certain constants S, which are 

 arbitrary excepting in so far as they are conditioned by the fact that the two 

 representations are to hold simultaneously for the function. The conditions to 

 which the constants S are subjected are given by the identity 



polynomial in {x, v) + ^J ((|, i;)). 

 This identity can immediately be replaced by another identity. 



h: 



5/*>f,(*'*)(.,«) ^ 4^ ^/^^f^*-^^") (^,^) _ 



(3) "V ^ ^'__\'',-«) + -^ 



polynomial in (s, 2>) + ^J ((^, »7))i, 



where the several functions ' ^ have a very simple connection with the corre- 

 sponding functions ^. The conditions to which the constants 8 are subjected 

 by this identity are obtained by equating to the coefficients of s~'' i>"^' 



(0 < )• <y + (« - 1) m, t = l,2, . . . n) 



on the left-hand side. The equations of condition so obtained are not all 

 independent. A linear equation between them with multipliers o,._] ,_i signifies 

 that the coefficient of s^' r""' in every product 



and in every product 



I 



is 0, where \ 



The interpretation of this is that \]r (;■, v) must have orders of coincidence which 

 are complementary adjoint to 



It) (A) 111 



-o-j, -a.,, . . . -o-^.^ 



for finite values z = a,,, and for the value z =■ o-. orders of coincidence which are 

 complementary adjoint to 



(cc) (o;) (CO) 



Tj — _, T^, — ^, . . . , Tj. — _. 



CO 



' For details in regard to these functions see the author's book Theory of the 

 Algebraic TunctUyns nf a Complex VarialjJe, Berlin: Mayer ife Mfiller. 



