ON ELLIPTIC AND HYPERELLIPTIC FUNCTIONS. 347 



From 



Section 7. — Two papers by Heine next come under our notice. The first of 

 these, " Untcrsuchuugeu iiber die Reihe 



(1 -./)(! -/) ^._^ (l-r/«)(l_5«+.)(l_^^)(l_^3 + r) „ 



+ (1_^X1_,/) (1-2X1-^X1-2^)0-?^+^) ' 



is in the thirtj--fourth vokime of Crelle's JournaJ. 



This series is denoted bv Heine by the symbols 0(a, /3, y) or <p(ix, /3, y, g, x), 

 as may be most convenient. I shall consider those parts of the paper which 

 relate to elliptic functions. Heine commences by showing- that the elliptic 

 functions 2Xr '^Kr 



oi-'x co*^ ^^ — - .)T- siu am - — - 



cos coam sm coam 



sm am . sm coam . • 



TT -TT IT ' 



— ' TTT!-, Sin coam 



2is..r IT TT 



7i siu coam 



IT 



can all be expressed by means of this series. 

 1 hus we imd —^ sm coam equivalent to 



l^W'1>a, i, h q\ -9€2'') + e-'>0(l, k, #, q\qe-^->^)], 



as is immediately seen by expanding the functions. 



Following the methods employed by Gauss for the hypergeometrical series, 

 Heine deduces a large number of equations, easy of ju-oof, of which I write 

 down the following : — ■ 



(/)(a, ft, y, q, x)—^{a., /3, y, q, q,v) -> 



(l_^yXl-^a+/3-vaOKa + l,/3,y)+^«+^-y,r(l-jV-3)d,(a + l,/3,y + l)-l 



-(l-'/-')Ka./3.7) = J ^"-^ 



2x2 



