212
A MEMOIR ON THE DOUBLE ^-FUNCTIONS.
[665
or finally it is
= 2l 0 ' - 2a 4 + a 3 (b + c + 2d + 2e + 2/) + a 2 {- (b + c)(d + e +f) - 2 (de + df+ ef)}
+ a {(b 4- c) (de + df+ ef) + 2 def) — (b + c) def
+ | a? — or (d + e 4- f) + o (de + df + ef) — def] (a + a 2 )
+ {- 4ci 2 + a (b + c 4- 2d 4- 2e + 2/) -bc-de — df— ef] aa,
+ 2a 2 a 2 2 .
It is to be observed that the investigation thus far has been entirely independent of
the values of SI/, S3', S': these values are, in fact, such as to make the coefficients
of (Sot) 2 , SotSu, (Su) 2 each equal to a constant, and it was really by such a condition
that the value of S(=S') was determined; but if we had thus also determined the
values of 2i 0 ' and S3', it would not have been apparent that the values of 2[ 0 ', 33'
and S' thus determined would be consistent with each other: the foregoing investi
gation of these values was therefore prefixed.
Completion of the reduction and final expression for AJ..
But now substituting the values of 2l 0 ', 33', S', we find
coeff. of (Sot) 2 = ab + ac + be + de + df+ ef
„ „ 2Sot du = — a 2 (a — b — c — d — e — f),
„ „ (du) 2 = — 2<x 4 + 2a 3 (b + c + d + e+f)
— a 2 (be + bd + be + bf+ cd + ce + cf+ de + df+ ef)
— (bede + bedf -f bcef+ bdef+ edef),
viz. these coefficients belong to the portion which contains the factor aa, of the
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expression for ^ ДA : the other portion was
(6' + c' + a + a,) d'ef (Sot) 2 — (V de) 2 (Sot) 2 — 2b' 9 и Sot — (a + а,) в' (du) 2 ,
where
b' = b'c'd'ef', b' = b —a, etc.
We have thus the complete result, viz. this is
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^ A A = aa 2 [(ab + ac+be + de + df+ ef) (Sot) 2
— a 2 (a— b — c — d — e—f) 2Эот 9и
Г— 2a 4 + 2a 3 (b + c + d + e+f) 'j
+ j — a 2 (be + bd + be + bf+ cd + ce + cf +de + df+ef) 1 (9г^) 2 ]
l - (bede + bedf + beef + bdef + edef) J
— (— 2a + b + с + а 4- а 2 ) (a — d) (a — e) (a —f) (Sot) 2 — (f de) 2 (Sot) 2
+ (a — b) (a — c) (a — d)(a — e) (a — f) 29w9ot
+ (a + a 2 ) (a — b)(a — c) (a — d)(a — e) (a — f) (da) 2 ,
which is obviously a sum of squares.
