697] 
ON THE DOUBLE ^-FUNCTIONS. 
427 
and the equation du = 0 then gives 
dx dy dz 
:=v+ / , , + -F== 
as it should do. The differentia] equation might also have been verified directly from 
any one of the expressions for 
Writing for shortness 
X = a - x .b — x. c — x .d — x, etc. 
then the general integral of the differential equation 
dx dy dz 
j — n 
VX + v'Y + x /Z ° 
by Abel’s theorem is 
x 2 , x, 1, \/X = 0, 
y\ y, 1, *JY 
z\ Z, 1, *JZ 
w 2 , w, 1, *JW 
where w is the constant of integration: and it is to be shown that the value of w 
which corresponds to the integral given in the present paper is w = a. Observe that 
writing in the determinant w = a, the determinant on putting therein x = a, would 
vanish whether z were or were not = y; but this is on account of an extraneous 
factor a — w, so that we do not thus prove the required theorem that (w being =a) 
we have y = z when x = a. 
An equivalent form of Abel’s integral is that there exist values A, B, G such 
that 
Am? + Bx + G = hJX, 
Ay 2 + By + G = *JY, 
Az 2 + Bz + G = \]Z, 
Aw 2 + Bw + C=\/lf, 
or, what is the same thing, that we have identically 
(Ad 2 + B0 + G) 2 - © = (A 2 - 1). 6 - x . 6 - y . 6 - z. 6 - w. 
or say 
C 2 — abed 
®y zw = ¿2 _ Y ’ 
54—2