ON ELLIPTIC AND HYPERELLIPTIC FUNCTIONS. 349 



Consequently putting /3=y=l in (1) and (6), we immediately obtain the 

 following equation : — 



0(a, 1, 1, q, x)= _— i^ ^(a, 1, 1, q, qx); 



1-x 



•'• ^^^' ^' ^' '^' ''^'^= %-t.'.'a-r-'^^? '^^'"' ^' ^' '^' *"'''^' 



and since q is siipposed less than unity, the series ^(a, 1,1, q, qKv) always 

 approaches unity as (h) indefinitely increases, and consequently 



(l-x){l-qx){l-q-x).... ^ ^ 



Again, putting .r=2^~"~^ in scries (2), wc have 



J. — jv 

 and repeating this process, 



^(a, A y, 2, 2V-a-^)= (l-2V-0.---(l-gv-^+"-) 

 ?K>l^7>1,^ ) (l_2y)....(l_2V+»-i) 



0(«-)-?i, /3, 7 + n, 2, 2^-''-^;. 



Now we have generally 



^(a + «,/3,y+", 2' -'*-') 



_, (l-^«+")(l-y^) (l-g.^+»)(l-^.^+«-i)(l.-g^)(l-gg+.) 



As («) increases without limit, this series approaches 



Hence when (n) is infinite, we have by (10) 



(1— Q'y-«)(1— 9v-»+i)- • . • 

 ^(a+n, /3, y + n, 3, gy-.-^) = ^^^_^^_^A^^ J^^_^_ /____^ 



and therefore 



^(a,/3,y,g,9V— P) 



_ (l- ^Y-'')(l-yY-a+') (l-gy-<^) ( l— f/Y-^+i). . . . 



— " (i_^v)(l— g'V+i) (1— 9V-«-^)(l— gy-«-^+)). 



From these formidaj it will readily be perceived that the leading properties 

 of functions ti can be deduced. For the details I refer to Heine's second 

 paper, " Abriss einer Theorie der elliptischen Functionen," which will be 

 found in the thirty-ninth volume of Crelle's Journal, and which, after the 

 remarks here made, will offer no difficulty. 



There are some consequences of formula (11) just proved which I shall 

 insert in this place. 



If we put 



^i' ' (l_2a+l)(l_g«+2)(l_2a+3^.... 



(11) 



