ON THE THEORY. OP NUMBERS. 



327 



00 



u= >J2ql II„ 



i+r 



oo 



«'=n„ 



(26) 



which are themselves perfectly determinate functions of to, if we understand 



17TCO 



the positive square root of 2 by \/2, and e 8 by 5-1. Of these functions, which 

 we shall designate by <j>(u>) and yp(to), the following equivalent expressions 

 have been given by Jacobi (Crelle's Journal, vol. xxxvii. p. 75-77) . 



u= V2q\ n 



1-q 4 



(l + q 2 »>- l )(l-q*">) 



"I 



,/ t T, TT ( l +g 4m " 2 )a-g 8m ) 



= */2qi 



o^H 



4nfl+2m 



J2m+m 



V 2 ^(°' M ) ; 



«= V22* n 



ft.o(o,|) 



(l-q 2 »>-*)(l—q im ) 



W27) 



(l — y*— 2 )^!— J*-) 



rTe, 



■•(^) 



-V9 7 ^(- 1 )"'g 2 '" 2+ '" — . 



u ■ L /2,v*n (1+g2 "'" 1)(1 - ym) 

 m- V 2g*n (1+2 2 M - l)2(1 _ 22 „ 



= sf2q 

 u' = U 



"71 o 0i0 (o, «) ; 



1 — J" 



(l + ^»-l)(l_ ? 2„ i) 



2( — l)j m (' n +i)«i(Sm s '-H>») 



(l+^-^'-'Xl— ? 4m ) 



_ n (l-^»- 1 ) 2 (l- 



H28) 



w'=n 



? 2»j\ 



(1— g 4 »- s ) a g.-i 2 «") 



_ s(-i)"y 2 = 9^(0,0,) 



S(-1)'Y'" 2 <V(0, 2«)' 



M ' =n (i-? 4m - 2 ) 2 (i-? 4 "') 



= ^-l)V» 2 _5 01 (0,2 w ) 



22" 



^o,o(0, w) 



