703] 
ON THE ADDITION OF THE DOUBLE ^-FUNCTIONS. 
461 
But the expressions 
aa 3 + fia 2 + <ya + S, ab 3 + fib 2 + 76 + 8, ac 3 + fie 2 + 7c 4- 8, 
— Va 2 — 6“ -4 12 . -434. 4 56 , Va- e 2 B 12 . T? 34 . i? 56 , Va 2 — e 2 (7 12 .6 34 . C 5f! , 
are 
respectively: the whole equation thus divides by 4 M . B x . C 56 ; throwing out this factor, 
VCL“ — 6 2 
and then multiplying each side by , we find 
V 0? — e 2 1 
v a t DE = —± - 
€ b — c.c—a.a — b 
Va 2 
b c . A 12 . A 34 . B x . C 56 
in which formula if we imagine 
Va 2 — e 2 
+ c a. B 12 • B^ . C 56 ,44.3g 
-\- a b. Cm • C34.4.56.B 56 }, 
v a? — e 2 v a? — e 2 
A u Tï u P 
e e 
each replaced by its value in terms of the xy- and ¿w-functions, we have an equation 
of the form 
Vet 2 — 
{x — z.x — w.y — z.y —w) DE m = 
X — z . x — w . y — z. y — w 
M, 
where M is a given rational and integral function of the 16 and 16 functions 
A12, AB U and 434, AB 3i of x and y and of z and w respectively. The factor 
Vet 2 - e 2 
(x — z.x —w.y -z.y — w) 
is retained on the left-hand side as being the same factor which enters into the 
equations for 4 56 , etc.: but on the right-hand side x —z.x —w.y —z.y—w should be 
expressed in terms of the xy- and ^¿¿»-functions. This can be done by means of the 
identity 
1, x+ y, xy 
x — z.x — w.y — z.y — w — 'fi 
1, x + y, xy 
1, Z + W, zw 
1, a + b, ab 
1, z + w, zw 
1, a + c, ac 
a —b. a —c 
where the summation refers to the three terms obtained by the cyclical interchange 
of the letters a, b, c. The first determinant, multiplied by a — b, is in fact 
— a — z. a — tv, a — x.a — y 
b — z .b —w, b — x .b — y 
and the second determinant, multiplied by a — c, is 
= j a — z.a — w, a — x.a — y , 
j c — z. c — w, c — x.c — y