663] FURTHER INVESTIGATIONS ON THE DOUBLE ^-FUNCTIONS. 
179 
We have moreover 
r r ,, 1 i abf.c'd'e' + a'bf'.cde] 
№ - M = (J ^y cdfMe . _ J f . abe \ 
-jr: (abc'd' — a'b'cd) (ef— ef), 
(x — x'y 
where for the moment a, b, a', etc., are written to denote a — x, b—x, a — x, etc.; we 
have then 
ef- e f = (e -«0(/- x)-(e- x)(/- x) 
= -{e -/) (x -x') = - ef(x - oc), 
and 
abc'd' — a'b'cd = (a — x)(b — x) (c — x) (d — x) — — (x — x) 1, x + x, xx' 
— {a — x') (b — x') (c — x) (d — x) 1, a 4- b, ab 
1, c + d, cd 
= — (x — x') [xx'abed]. 
Hence [ab] - [cd] = ef[xx'abcd], and the equation to be verified becomes 
(ef. cd — cd. ef) [xx'abed] = 0, 
viz. this is, in fact, an identity. 
Cambridge, 14 March, 1877. 
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