422 
[697 
697. 
ON THE DOUBLE ^-FUNCTIONS. 
[From the Journal für die reine und angewandte Mathematik (Crelle), t. lxxxvii. (1878), 
pp. 74—81.] 
I have sought to obtain, in forms which may be useful in regard to the theory 
of the double ^-functions, the integral of the elliptic differential equation 
dv 
dx 
= 0: 
fa — x.h — x.c 
the present paper has immediate reference only to this differential equation; but, on 
account of the design of the investigation, I have entitled it as above. 
We may for the general integral of the above equation take a particular integral 
of the equation 
dx dy dz 
fa — x.h — x.c—x.d — x fa — y .h — y. c — y. d — y fa — z .h — z. c — z. d — z 
viz. this particular integral, regarding therein 0 as an arbitrary constant, will be the 
general integral of the first mentioned equation. And we may further assume that z 
is the value of y corresponding to the value a of x. 
I write for shortness 
a — x, h — x, c — x, d — x = a , b, c, d, 
a-y, h-y, c-y, d-y = & l , b 1} c 1? d x ; 
and I write also (xy, he, ad), or more shortly (he, ad) to denote the determinant 
1, x + y, xy 
1, b + c, he 
1, a + d, ad 
we have of course (ad, hc)=—(hc, ad), and there are thus the three distinct determinants 
(ad, he), (hd, ac) and (cd, ah).