458 ON THE ADDITION OF THE DOUBLE ^-FUNCTIONS, 
or as it may be written 
[703 
fi .X — Z .X — w ,y — z.y — IV = 
e.a — z.a — w 
x-y 
{y — z. y — w. a — y.'dX — x — z. x — iu. a — x.^ Y) 
€ • a — X . & y f / fy / Tiri 
H \w — x.w — y.a — w. VZ —z — x.z — y.a — z.v vv \, 
z — W 
an equation for the determination of O. 
Consider first the expression which multiplies e.a — z.a — w, this is 
= 7r{y- z -y 
C'lQ 
we have 
— w. a 2 VX — x — z . x — w . a 2 
BE l2 = J- {Vbie 1 f 1 a 2 c 2 d 1 , — Vbaeof^ajCidj], 
and multiplying this by 
we derive 
-d]2 • Cj2 • Di%, — Va^d^a/h, 
BE l2 . C 12 . D 12 . A 12 = ~ {c 2 d 2 a 2 VX - Cjd^ V Y}, 
t/ 12 
and similarly two other equations; the system may be written 
BE. C. D. A — ~ {c 2 d 2 a 2 VX — c^aj V Y], 
t/j2 
GE .D.B. A — „ {d 2 b 2 „ „ - djbj „ „ }, 
DE. B. C. A = „ {b 2 c 2 „ „ -bac 1} , „ }, 
the suffixes on the left-hand side being always 12. The letters b, c, d which enter 
cyclically into these equations are any three of the five letters other than a; the 
remaining two letters e and f enter symmetrically, for BE is a mere abbreviation for 
the double triad BEF. AGD; and the like for CE, and DE. 
Multiplying these equations by 
b — z.b — w c — z.c — w d — z.d—w 
b — c. b — d’ c — d. c — b’ d — b.d — c* 
respectively, and then adding, the right-hand side becomes 
= \y —z .y — w . a 2 y'X — x — z. x — w. a, V F}. 
b — z. b — w —1 
b — c.b — d c — d.d — b.b — c 
. c — d . B 3l 2 , etc., 
W riting