208 
A MEMOIR ON THE DOUBLE ^-FUNCTIONS. 
[665 
The formula thus becomes 
,{s-£p-gp.} 
— (3w + 0 duf — ^ (dvr - 6 duf + i (abcd^fj + a^c^ef) (0ot) 2 
— (V (¿e) 2 (0ot) 2 . 
The second line here is 
— {a, 1 U + a Uj) (0w) 2 — -q (a x U — a t/j) 0m 0ot 
+ ^ {— a ! £/" — a U l + abcdjejb 4- aJbjCjdef} (0w) 2 , 
and the coefficient herein of (0ct) 2 is = (ad^jfj — ajdef) (be — biC x ). Writing for the 
V“ 
moment d —a, e — a, f — a = d', e, f, we have 
\ (adjejfj — a x def) = {a (d' + a x . e + a x ./' + a x ) — a! {d' + a. e + a4- a)} 
u a x — a 
= — d'ef 4- aa x (d' 4- e +/' + a + a x ), 
в 
(be — biCi) = — (&' 4- d 4- a 4- a x ). 
The whole term in (0ет) 2 is thus 
= {(b' 4- c + a 4- a x ) d'e'f' + aa x (b' + c' + a 4- a x ) (d' + e' +f' + a + a x )} (0ет) 2 . 
2 
The coefficient of ди дът is — ^ (a x U — a f): viz. this is 
2 , 
8/j 8i 
and if 
{a x (b' + a. c' + a. d' 4- a. e' 4- a+ a) — a (b' 4- a x . c 4- a x . d' + a x . e 4- a x ./'+ a x )}, 
that is, 
this is 
b' 4- a . c 4- a. d' 4- a. e + a./' 4- a = b' 4- c'a + D'a 2 4- E'a 3 + F'a 4 + a 5 , 
b' = b’c d'e'f, c' = Ib'c'd'e, d' = Ib'c'd', e' = 26V, 
f' = Xb' = b' + c' + d' + d + /', 
= 2 |b' — D'aa x — Eaaj (a + a x ) — F'aa x (a 2 4- aa x 4- a x 2 ) — aa x (a 3 4- a 2 a x + aa x 2 + a x 3 )} : 
or say for shortness it is = — 2 (b' — aa^) where 
= r/ + e' (a + a x ) 4- f' (a 2 + aa x + a x 2 ) + a 3 + a 2 a x 4- aa x 2 4- a x 3 ; 
the term in question thus is — 2 (b' — aa x <ï>) du dm. 
The coefficient of {du) 2 is — (af + a CT^), viz. this is 
— a x (b' + a . c + a. d' + a . e + a f + a) — a (b' + a x . c + a x . d' + a x . e' + a x . f + a x ), 
which is = — (a + a x ) b' — aa^F, where 
Ÿ = 2c' + d' (a + a x ) + e' (a 2 + a x 2 ) + f' (a 3 4- a x 3 ) + a 4 + a x 4 .