[722 
groups, but it 
723] 
63 
i-commutative 
) made up of 
a?ßa 2 ß 2 ; and 
commutative, 
723. 
VARIOUS NOTES. 
then 
[From the Messenger of Mathematics, vol. vm. (1879), pp. 45—46, 126, 127.] 
An Algebraical Identity: p. 45. 
Let a, b, c, f g, li be the differences of four quantities a, /3, 7, 8, say 
a, b, c, f g, h = /3-ry, 7 - a, a - /3, a-8, (3-8, 7 - 8; 
h — g + a = 0, 
— h . + f+ 5 = 0, 
g -f . + c = 0, 
— a — b — c .=0. 
Now Cauchy’s identity 
(a + b) 7 — a7 — b 7 = lab {a + b) (a 2 + ab + ¥f, 
putting therein a + b — — c, becomes 
a 7 + b 7 + c 7 = 7abc ( be + ca + ab) 2 ; 
hence we have 
h 7 — g 7 + a 7 — — 7agh {— ga + ah — hg"f, 
— h 7 . +f 7 +b 7 — — 7 bhf (— hb + bf —fhy, 
g 7 -f 7 ■ + c 7 = - lefg (-fc + eg -gf) 2 , 
— a 7 — b 7 — c 7 . = — 7 abc ( be + ca + ab) 2 ; 
whence, adding, 
agh (- ga + ah - hgf + bhf {- hb + bf—fh) 2 + cfg {-fc + eg- gf) 2 + abc {be + ca + ab) 2 = 0,