326 Mr Baker, On a certain automorphic function. [May 27, 



where r,;^ j ... , n^p are the, definite, periods of Vi, and 



fCi) . . . , aJ^j , A,i , . . . , A.^ 



are rational integers ; the values of these integers are quite definite 

 when the barriers, spoken of, are drawn. 



Hence the function 



F(n= ^(^> 'in)B(^, on) ^i (x^j,/, »«+...+ x^^,/, m^ 



^^^ @ {vf"^ - vi"^^' '^i - ... - Ui™/- 'V) 



has no poles, and no zeros, and, by the properties of the function, 

 is such that 



Fa,,)IF{^) = {-y>^+K-K 



Thus the square of i''(^) is a constant ; thus F (^) is a constant, 

 and 



where K^, ..., Kp are unknown rational integers; and 

 X{t m)E(^, m) 



=4,-«(v/''"+...+v/'-)@r ,,._M^_i(^^ 



where A is independent of ^. 



By taking ground the circle C^, and considering the factors of 

 the two sides, we can infer that Xn is even, = 2Ln say ; and can 

 then write 



X(r, 7n)E(^, m) 



_ ^ g - TTiSL^ (^„ + K) - TTi (TaLi^ + . . . + 2ri2LiL2 + • • • ) e [y/- '« - 1 (^^ + hi)\ 



where L^, ..., Z^ are integers depending on m and h^, ..., Ap, and 

 A does not depend on ^. 



But, using the equation 



\(^, m) = -X(m, ^), 

 we can infer that this equation is equivalent to 



X (^, m) E iX, m) = C@ [vf ^-h{9i+ fh)], 

 where C does not depend on ^ or m. 



Hence, putting ^ = m, using the equations 

 [^(r, m)/(r-m)]^.,, = l, 

 [X(^, m).(^-7?i)]^=,„ = l. 



