665] 
A MEMOIR ON THE DOUBLE ^-FUNCTIONS. 
209 
The formula thus is 
fp A4 = aaj js — P — ^.Pi — (b + c'+ a + aj)^'-}- e + f'+ a + a : ) (0ct) 2 + 2<T> du dm — ”'F (0w) 2 | 
— (a + a a ) b' (idu) 2 — 2b' 3m 0ot + (b' + c' + a + a x ) rf'e/' (dm) 2 — (Vdef (dm)-. 
The whole coefficient of aa l5 substituting for 2, P, P u <ï>, Ÿ their values, and arranging 
according to dm, du, is 
( 9 77 TV / 9 TJ 77 \ i 
= (3®) 2 [S' + -^r - 0T + (- y - ^ j - (P + c' + a + sf) (d' + e' +/' + a + aOJ 
a /2ff ( 2Z7i Z7A tT ^ 
+ ^ 0tJT 0M |S3 + a, ^ j + a ( #3 5)2 ) 02 02 
+ d' + e' (a + ad + f' (a 2 + aaj + a/) + a 3 + a 2 aj + aa/ + a/j 
+ (a») 2 j«/ + v ( 2 ''- ?) + a 2 (-^ + (a -2a,)^ 
— ^ (abcd^jf, + a,b,Cidef ) — 2c' — 1/ (a + a,) — E' (a 3 + a, 3 ) — F' (a s + a, 3 ) — a 4 — a, 4 | : 
and we have to reduce separately the three coefficients of this formula. 
Third step of the reduction. 
First, for the coefficient of (0-53-) 2 ; recollecting that 6 — a! — a, we have 
2 —^— 1 = - 2c' — 2d' (a + a 2 ) — 2e' (a 2 + aa! + a/) — 2f' (a 3 + a 2 aj + aa/ + a/) 
6 
— 2 (a 4 + a 3 a, + a 2 a/ + aa/ + a/), 
~(U'+ Ui) = 2c' + 2d' (a + aj) + 3e'(a 2 4-a/) + 4f' (a 3 + a/) + 5 (a 4 + a/). 
Adding these, the right-hand side divides by (aj-a) 2 , that is, by 6 2 ; and the resulting 
value is 
= e' + 2f' (a + aj) + 3 a 2 + 4aaj 4- 3a/. 
The term - (6' + c' + a + af)(d' 4- e' +/' 4- a 4- a^, attending to the values of e' and f', is 
= h'c' + d'e' 4- d'f 4- ef - e' - f' (a 4- a^ - a 2 - 2aaj - a/ ; 
hence the whole coefficient of (dm) 2 is 
= QT' + h'c' 4- d'e 4- d'f + ef'' 4- f' (a 4- a x ) - 2 (a 2 + aaj 4- a/), 
or substituting for h', c, d!, e, f their values, this is 
= (S' + 4a 2 - a (h + c + 2d + 2e + 2/) + he + de + df+ ef+ (f - 6a) (a + a,) - 2 (a 2 4- aa a + a/). 
Proceeding next to reduce the coefficient of 2 0ct0w, observing as befoie that 6 — q,i a, 
we have 
2a 1 £T-2a?7 1 = 2fi , _ 2l y aai - 2E , aa 1 (a + a x ) - 2Faa ] (a 2 4- aa, + a/) - 2aaj (a 3 4- a 2 ^ 4- aa/ + a/), 
27 
c. x.