210 
A MEMOIR ON THE DOUBLE ^-FUNCTIONS. 
[665 
also 
-(a 1 ET + ï7)-(aüÏ4- U x ) = 
— 2b' 4- d' (— a 2 4- 4aaj — a^) 4- e' (— a 3 + 3a 2 ai 4- 3aaj 2 — a^) 4- f' (— a 4 + 4a 3 a, 4- 4aai 3 — a x 4 ) 
— a 5 4- 5a 4 aj 4- 5aap — a x 5 ; 
adding these two expressions, the right-hand side divides by (aj — a) 2 , that is, by 9' 2 , and 
the resulting value is 
= — d' — e' (a + a 3 ) — f' (a 2 4- a x 2 ) — a 3 + a 2 aj 4- aa a 2 — a^. 
To this is to be added 
4- 2d' + e' (a + a x ) 4- f' (a 2 + aa x -f a^) + a 3 + a 2 a z -t- aa^ + a x 3 ; 
we thus see that the whole coefficient of 23u 3sr is 
= d' + Faa x 4- 2 (a 2 aj 4- aa, 2 ), 
or say it is 
= d' + (f — 6a) aa! + 2 (a 2 a x + aa^). 
Lastly, for the coefficient of (du) 2 ; we have 
2a! 2 27 - 2a 2 C/! 
6 
= 2b' (a 4- a x ) 4- 2c'aa! — 2E'a 2 a! 2 — 2F'a 2 a! 2 (a 4- a^ — 2a 2 a! 2 (a 2 4- aaj + a x 2 ), 
and also 
- a 2 U' + (a - 2aj) Ü - a 2 U,' + (a, - 2a) U = 
— b' (a 4- a a ) 4- c' (2a 2 — 4aaj 4- 2a! 2 ) + d' (a 3 4- aj 3 ) 4- e' (a 4 — 2a 3 a! 4- 6a 2 a! 2 — 2aaj 3 4- a^) 
4- f' (a 5 — 2a%i 4- 4a 3 ai 2 4- 4a 2 a x 3 — 2aaj 4 4- ai 5 ) 4- (a 6 — 2a 5 a, 4- 5a 4 a, 2 4- oa 2 aj 4 — 2aa! 5 4- a/), 
whence the sum of these two expressions is 
= b' (a 4- aj) 4- c' (2a 2 - 2aa x 4- 2ai 2 ) 4- d' (a 3 4- a^) 4- e' (a 4 — 2a 3 aj 4- 4a 2 ai 2 — 2aaj 3 4- aj 4 ) 
4- f' (a 5 — 2a 4 a! 4- 2a 3 ai 2 4- 2a 2 a! 3 — 2aai 4 4- ai 5 ) 4- a 6 — 2a 5 ai 4- Sa.% 2 — 2a 3 a 1 3 4- 3a 2 ai 4 — 2aa 1 5 4- ai 6 . 
We must to this add the term — (abcdie^ 4- ajbjCidef), that is, 
— a (£>' 4- a. c' 4- a. dl 4- %. e + aj4- aj) — ai (&' 4- &i. c 4- a x . d' 4- a . e 4- a ./' 4- a). 
Putting for the moment 
d! 4- a. e' 4- a .f 4- a = t 4- cr'a 4- p'a 2 4- a 3 , 
that is, 
r - d'ef, a = d'e' 4- df 4- ef, p' = d' + e'+f, 
the term is 
— b'c'r (a 4- a!) — (b' 4- c) r (a 2 4- a x 2 ) — t (a 3 4- a/) — (b'c' 4- a') (a 3 a x 4- aaj 3 ) 
— (b' 4- c' 4- p) (a 3 ai 2 4- a 2 a x 3 ) — 2a 3 ai 3 
— 23'cVaai — [(6' 4- c) a 4- b'c'p'] (a 2 a 2 4- aaj 2 ) — 2 (&' 4- c') pa 2 ai 2 .