703] ON THE ADDITION OF THE DOUBLE ^-FUNCTIONS, 
the left-hand side becomes 
459 
= c ^jZIJ~ C i c ~ d - R * • BE * • G 12 .D K + d—b. G 3i 2 . CE V1 .D 12 .B 12 +b-c. D u >. DE n . B a . C 12 ], 
which for shortness may be written 
Aio 
c — d . d — b .b — c 
£ {c — d. Bu 2 . BE 12 . C 12 . D 12 ], 
the summation referring to the three terms obtained by the cyclical interchange of 
the letters b, c, d. The result thus is 
■w-{y — z .y — w. a 2 VZ - x — z.x — w.aj V F} 
C'19 
-A. 
c—d.d — b.b — c 
£ {c — d. B M 2 . BE 12 . C 12 . D 12 \. 
Interchanging x, y with z, w respectively, we have of course to interchange the suffixes 
1, 2 and 3, 4; we thus find 
x~{w — x ,w — y Z — z— x. z — y. a 3 V W] 
V'U 
-A, 
c- d. d— b .b — g 
%{c — d. B x f. BE U . . D34}, 
and we hence find the value of fl.x—z.x — w.y — z.y — w. But O, = aa 3 + /3a 2 + ya + 8, 
is = va ! - e s . A 12 . Am . A S6 : the resulting equation divides by A 12 .A U : throwing out this 
factor, we have 
VO? — £~ 
(x — z . x — w ,y — z .y — w)(c — d.d — b.b — c) A x 
€ 
= A s 4 £ {c — d . B 3i 2 . BE 12 . C 12 . A,} + A 12 £ {c — d . B 12 -. BE34. C34. A-il, 
where, as before, the summations refer to the three terms obtained by the cyclical 
interchange of the letters b, c, d; these being any three of the five letters other 
than a; and the remaining two letters e, f enter into the formula symmetrically. 
The formula gives thus for A 56 ten values which are of course equal to each other. 
Writing for a each letter in succession, we obtain formulae for each of the six 
single-letter functions A x of p and q; and the factor 
V a 2 — e 2 
{x — z.x — w.y — z.y — w) 
is the same in all the formulae. 
We require further the expressions for the double-letter functions of p, q. Con 
sidering for example the function DE 56 , which is 
= t, 1 {Vd 5 e 5 f 5 a 6 b 6 c 6 - Vd 6 e 6 f 6 a 5 b 5 c 5 },