211 
665] A MEMOIR ON THE DOUBLE ^-FUNCTIONS. 
Adding it to the preceding expression, the sum is 
= 0' - b'c'r') (a + a,) + {2c' - (6' 4- c') r] (a 2 + a, 2 ) + (d' r r') (a 3 + a*) + e' (a 4 + a x 4 ) 
+ {— 2c' — 2b'c'a'} aa x — {(b’ + c) a + b'c'p'} (a 2 a x + aa x 2 ) - (2e' + b'c' 4- a) (a 3 aj 4- aa x 3 ) 
4- {4e' - 2 (b' 4- c') p'} aV 
4- f' (a 5 + a x 5 ) + a 6 + a x 6 
— 2f' (a 4 a x + aa x 4 ) — 2 (a 5 a x + aa x 5 ) 
+ (2f ' -b' — c' — p) (a 3 a x 2 4- a 2 a x 3 ) + 3 (a 4 a x 2 4- a 2 a x 4 ) — 4a 3 a x 3 . 
This is, in fact, divisible by (aj — a) 2 , that is, by 6-: for we have between the symbols the 
relations 
F ' = b' + c' + p', 
e = b c (b 4- c ) p + (j, 
d' = b'c'p' + (b' 4- c‘) a' + t, 
c' = b'c'a' + (b' + c') T, 
b' = b’c'r', 
and we thus reduce the expression to 
{2c' — (6' + c') r'} (a 2 — 2aa x + a x 2 ) + (d' — r') (a 3 — a 2 a x — aa x 2 4- a x 3 ) 
+ e' (a 4 — 3a 3 a x + 4a 2 a x 2 — 3aa x 3 + a x 4 ) 4- (b' + c) p' (a 3 a, — 2a 2 a x 2 + aa x 3 ) 
4- f' (a 5 — 2a 4 a x + a 3 ax 2 4- a 2 a x 3 — 2aa x 4 4- a x 5 ) 
+ (a 6 — 2a r, a 1 4- 3a 4 a x 2 — 4a 3 a x 3 4- 3a 2 aj 4 — 2aa x 5 4- a! 6 ), 
viz. effecting the division, the quotient is 
= 2c' - (b' 4- c') r 4- (D' - t) (a 4- a x ) 4- e' (a 2 4- a x 2 ) 4- f' (a 3 4- a x 3 ) 4- a 4 4- 2a 2 a x 2 4- a x 4 - (b'c' 4- a') aa x . 
To this must be added 
- 2c' - d' (a 4- a x ) - e' (a 2 4- a x 2 ) - f' (a 3 4- a x 3 ) - (a 4 4- a, 4 ) ; 
and we thus obtain the coefficient of (3iCf in the form 
8l 0 / — (b‘' 4- c) t' — r (a 4- a x ) — (b'c' 4- cr ) aa x 4- 2a 2 a x 2 , 
viz. this is 
= 2l 0 ' 4- (b + c - 2a) (a -d)(a- e) (a -/) 4-(a-d)(a- e) (a -/) (a + a x ) 
4- {- (a - b) (a-c)-(a- d) (a-e)-(a- d) (a -/) - (a - e) (a -/)} aa x 4- 2a 2 a x 2 , 
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