1879.J the Values of the Theta and Zeta Functions. 353 

 where 6 is the modular angle, it follows that 



e 3 (|K+^K , )=?-* — W*(ooa \6 + i sin $0), 



7T 



and a(JK+ JiK 1 ) = ? -^^W*(cos j0+i sin £0). 



This last result is readily identified with that found above, since 



cos J0= J_211_Z_^L, sm J0=-J 



§ 3. From the values of Qu for u=%K, ^'K', ^K-f^iK', the values 

 of Qu for the other arguments contained in the table were deduced by 

 means of the formulae 



e(u + K) =7c , ~ i dnuQu, 



iiru 



Q(u -f- iK') =iq~ i h i e~W sn u Qu. 

 The values of ~Su were deduced from those of Qu by the formula 



iiru 



Hu= —iq^evKQ (u + ?K') . 



For example, 



H(JK + ^K') = -z^ (iK+iiK,) eftK+fiKO 



= — iq^e^cf . q~^^.Jc i k /i (sui + i cos ^6>) 



= 2 -A — i) (sin \6 + i cos J0) 



= 2 -TV?!5.W4cosK^-^)+^sini( 7 r-^)} ; 



and it can be shown that 



cos J(tt-0) =i ^ — ^J-, sm i(V-0) =2 A_ Ll_. 



The values in the table were verified throughout by the formulae 



B.u 7 x H(w+K) U 



Qu Qu h' k 



several of the results are of course deducible at once from one another 

 by the relations 6(?t + 2K) = 9%, H(w + 2K) = — Hw, and a change in 

 the sign of i. 



