208 Doubly Periodic Functions. [Mar. 12> 



: jo ■ / x — 



1 



f r ^ n ^ 1 



where sm0— 



yi-^ 2 sin 2 



3. Remarks on a function, which includes Jacobi's Z(V) as a 

 particular case. 



If we regard p as a function of u and write — — = «, there seems 

 & x n K 



to be a V(u) relation of the form V(u) = < I . 



This satisfies the two equations 



g(u + 2iK') — g(u) = — ~, as may be seen by changing (1) 



u, p into u + 2K, p-\-2ll', (2) u, p into u + 2iK', _p-f 2^P', respec- 

 tively. 



It is easy to show by ordinary trigonometrical formulas that we get 

 a solution of these functional equations by putting 



rW = ^(r--o smw sin^ + J-^-sin2i* s: 

 M ft K ll-r K 1— O 4 



. 2ttu 



K ' *l- 2 4 u ~ m ^ 



1-2 



where u is a constant and q = e~ 7T \. 



^ sin3^ sin^i+ .... adinfin. 1 ^ 

 1— o b K J 



The deduction of- 



E 2tt { q 



. iru , q 2 . 2ml 

 ■ sin — + ^ — - sin — — . 

 K 1— 2 4 K 



presents no particular difficulty. 

 This is Jacob's Z(u). 



The above considerations suggest the very interesting question 

 whether we may not take u to be a function of p, and so write 



p — ILu= A l sin !^-j-A 3 sin ?^ + A 3 sin %Z -f . . . acZ tn/m. 



K ii n n 



Prima facie this does not seem to be an impossible equation, since 

 i.lie series can be reversed by Lagrange's theorem, i.e., we may have 



p — — u=B 1 sm — + B 3 sm _ - + . . . . 

 K 1 K K 



