432 
[699 
699. 
ON THE TRIPLE ^-FUNCTIONS. 
[From the Journal fur die reine und angewandte Mathematik (Crelle), t. lxxxvii. (1878), 
pp. 134—138.] 
There should be in all 64 functions proportional to irrational algebraical functions 
of three independent variables x, y, z\ there is no difficulty in obtaining the expression 
of these 64 functions in the case of the system of differential equations connected 
with the integral 
jdx : fa — x.b — x.c — x.d — x.e — x.f — x.g — x.h — x\ 
but this is not the general form of the system for the deficiency (Geschlecht) p = 3; 
and I do not know how to deal with the general form: the present note relates 
therefore exclusively to the above-mentioned hyper-elliptic form. 
I. 
If in the Memoir, Weierstrass, “Theorie der Abel’schen Functionen,” Crelle, t. LII. 
(1856), pp. 285—380, we take p = 3, and write x, y, z; u, v, w; a, b, c, d, e, f g 
instead of x x , x 2 , x s ; u 1} u 2 , u 3 ; a x , a 2} a 3 , a i} a 5 , a 6 , a 7 \ then, neglecting throughout 
mere constant factors, we have 
X — a — x .b — x . c — x. d — x. e — x .f— x . g — x, 
with the like values for Y and Z: the differential equations are 
7 b — x.c — x.dx b — y.c — y.dy b — z. c — z.dz 
du = VX + VY + V2 
7 c — x. a — x.dx c — y .a —y .dy c —z .a — z .dz 
*-—vx—+ ~xr + —yz— ■ 
_a — x.b — x.dx t a — y.b — y.dy + a — z.b — z.dz 
dw = 
JX 
+ 
fY 
JZ