BIRKHOFF. — THE GENERALIZED RIEMANN PROBLEM. 5G3 



We have also seen that 



(61) Vii{t+ 1) = Vi}(i)- 



Now let us attempt to satisfy (60) and (61) by writing 



(62) pij (0 = ce«''+*'o- {t — ai)...<y{t — a J, 



where cr{i) is the Weierstrass sigma function belonging to the periods 



, 27rV:ri; 



OJ = 1, CO = — , , 



logg 

 which satisfies the relations 



(63) 



log? / 



I logg 



The above choice of w and w' meets the requirement 9t (w'/co) > 0, 

 since \q\ > 1. 



A direct substitution into (61) determines 



(64) a =—7?^, 6 = r7X«i+ MttV— 1 + 2^-7rV— 1, 



1=1 



in which k denotes any integer. If these values of a and b are taken 

 and a direct substitution is made in (60), there is obtained the further 

 condition 



(65) 2- «i H ] ^ (^i -r Pj -r I — 



i=l 



log q log q 



in which I denotes an integer. These conditions are necessary and 

 sufficient that an expression of the form (62) shall satisfy both (60) 



and (61). If the value for Y^ai deduced from (65) be used in (64) 



1=1 

 the expression for h simplifies to 



But if one adds or subtracts a period to aj the precise effect is to alter 

 k or I by an integer. It is therefore always permissible to take k = 1 = 

 and to write 



(66) pijit) = cue 2 -+1''"^+'^^ ■■^i'ait- a^^'^^) . . . a (t - a,^^^'), 



