703] 
455 
703. 
ON THE ADDITION OF THE DOUBLE ^-FUNCTIONS. 
[From the Journal für die reine und angewandte Mathematik (Crelle), t. lxxxyiii. (1879), 
pp. 74—81.] 
I assume in general 
% = a—6.b — O.c — 6.d—6.e — 6.f — 6, 
and I consider the variables x, y, z, w, p, q, connected by the equations 
1, 
1, 
1, 
1, 
L 
l 
X, 
y> 
w, 
P> 
<1 
X 2 , 
y\ 
s 2 , 
IV 2 , 
p\ 
q 2 
QQ'3 
y s , 
p, 
w 3 , 
p 3 , 
q 3 
VX, 
Vr, 
VI, 
fw, 
VP, 
■JQ 
equivalent to two independent equations, which in fact serve to determine p, q, or say 
the symmetrical functions p + q and pq, in terms of x, y, z, w. 
These equations, it is well known, constitute a particular integral of the differential 
equations 
dx dy dz dw dp dq _ .. 
Vx + Vf + VI + VW + VP + Vq ~ ’ 
xdx y dy zdz w dw p dp q dq 
Vx + VF + Vf + VW + Vp + Vq 
= o, 
or what is the same thing, regarding p, q as arbitrary constants, they constitute the 
general integral of the differential equations 
dx dy dz dw _ 
Vx vy fz Vw 
xdx y dy z dz w dw ..