684] 
293 
684. 
ON A RELATION BETWEEN CERTAIN PRODUCTS OF 
DIFFERENCES. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xv. (1878), 
pp. 174, 175.] 
Consider the function 
1 abc . de 
' abd . ce ' 
+ bed . ea 
+ bee . da 
^ + ede . ab 
► < 
+ eda . eb V 
+ clea. be 
+ deb . ac 
+ eab . cd 
+ eac . bd 
where 
abc = (a — b) (b — c) (c - a), 
ab — (a — b) (b — a), = — (a — b) 2 , 
&c. ; 
therefore 
abc = bca = cab = — bac, &c.; 
ab = ba. 
It is to be shown that the function vanishes if e = d. Writing e = d, the value is 
3 (bed. d,a + dab . cd) — abd . cd 
— bed . da 
— eda . db 
— dac . bd,