462 
ON THE ADDITION OF THE DOUBLE ^-FUNCTIONS. 
[703 
so that the formula may also be written 
— on = y 
a — z. a — io, 
a —x .a 
-y 
\ a — z.a- 
-w, 
a 
— x.a — y 
V 
b — z .b —w, 
b —x .b 
-y 
| c — z. c - 
-w, 
c 
1 
Si 
cs 
1 
* 
(a —b) 2 (a— cy 
x — z.x — w.y — z.y — w 
or, what is the same thing, it is 
v - A V ?B,?) (A^C V ? - A v ;-a/) 
X — Z .X - w .y — z.y — W — 2. -— ~ ——, 
9 9 (a — by (a — c) I 2 
which is the required expression for x — z.x — w.y —z.y —w; the letters a, b, c, which 
enter into the formula, are any three of the six letters. 
As regards the verification of the identity, observe that it may be written 
v [L + M(a + b) + Nab] {L + M (a+ c) + Nac} 
x — z.x —w.v — z.y—rv = z - - i, 
9 J a-b.a-c 
where L, M, N are 
= (x + y) zw — (z + w) xy, xy — zw, and z + w—x — y: 
this is readily reduced to 
x-z.x — w.y — z.y — w= M- — NL, 
which can be at once verified. 
Cambridge, 12th March, 1879. 
I take the opportunity of remarking that, in the double-letter formulae, the sign 
of the second term is, not as I have in general written it —, but is +, 
AB = 
1 
x-y 
{v abfCidjej + V a^^cde}, etc. 
In fact, introducing a factor &> which is a function of x and y, the odd and even 
A-functions are =6oVaa 1 , etc., and 
—— {Vabfc 1 d 1 e 1 + Vajbdicde}, etc., 
x — y 
respectively; ® is a function which on the interchange of x, y changes only its sign; 
and this being so, then when x and y are interchanged, each single-letter function 
changes its sign, and each double-letter function remains unaltered. 
Cambridge, 29tli Jidy, 1879.