294
ON A RELATION BETWEEN CERTAIN PRODUCTS OF DIFFERENCES. [684
viz. this is
3 bed . ad — abd . cd
+ 3 abd. cd — bed . ad
— 2acd . bd
= 2 bed . ad — 2acd . bd 4- 2abd. cd
= 2 (bed . ad + cad . bd + abd . cd),
which is easily seen to vanish; the value is
(b — c)(c — d) id — b) (a — d) 2 = — (b — c) (a - d) 2 (b — d) (c — d)
+ (c — a) (a — d) (d — c) (b — df — (c — a) (a — d) {b — d) 2 (c — d)
+ (a — b)(b — d) (d — a) (c — d) 2 — (a — b){u — d) (b — d) (c — cIf:
viz. omitting the factor {a — d)(b — d) (c — d), this is
= — (b — c) (a — d)
— (c - a) (b — d)
— (a — b)(c— d),
which vanishes. Hence the function also vanishes if e = a, or a = b or b = c, or c = d;
and it is thus a mere numerical multiple of (a — b)(b — c) (c — d){d — e) (e — a), or say it
is = Mabcde.
To find M write e = c, the equation becomes
3abc . dc — eda. cb = Mabcdc, — Mabc . dc,
+ 3bed. ca — ac
+ 3 dca. be
+ Scab . cd,
viz. this is
6abc . dc + 4dbc. ac + 4adc ,bc = M. abc. dc,
giving M = 10. In fact, we then have
— 4abc . dc + 4dbc . ac + 4adc .bc = 0,
that is,
which is right.
— abc. dc — bde . ac — dac .bc = 0,
And we have thus the identity
' abc . de 1
' abd . ce '
+ bed . ea
+ bee . da
V + ede . ab
> <
4- eda . eb v
+ dea. be
4- deb . ac
+ eab . cd
4- eac . bd
= 10 . abede,
or say
3 [abode] — [acebd] = 10 [abode].