42 



Art. 7. — K. Tera-îawa; 



accomplished Ijy using tlie quantity qi, instead of y, wiiich is 

 defined by 



g, = e-"^i, 



Now, by the aid of the relation 



ê[{o) ä, ß, &, 



the expression of èe/«;,— --y, found in §24 may be transformed into 



Making use of the transformation formulae of Theta-functions it 

 will he easily shown that 



^o'(o/t) _^. ,.., ß:/(o/T,) 



'>o(o/r) 



^,(o/rJ '■ 





'UolT) - ' Hoir,) ' 



consequently we liave 



-- L I J, loir,) &loiT,) J J 



The </rseries for tlie functions needed here are as follows: 



^., =z 3gi(l-|-g,- + gi' + g;'-'+ ), 



.>;' = - \i7i\q, + 2--g,^ + '^'q,' + ), 



d,= l + 2g, + 2g,'' + 2g:«+ , 



/A, = l_2g, + 2g,^-2g,"+ 



The quantity q^ will be found from 



