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].