viz. substituting for a, 6, c their values g — h, h—f f— g, this is an identical equation. 
ON AN IDENTICAL EQUATION CONNECTED WITH THE THEORY 
OF INVARIANTS. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xn. (1873), 
pp. 115—118.] 
Write 
a = g-h, 
b=h-f 
c = f-g, 
equations implying a fourth equation forming with them the system 
. — h + g — a — 0, 
h . -/-6=0, 
-g+f . - C = 0, 
a+b + c . =0, 
and also 
af+ bg + ch = 0. 
Then, putting for shortness 
P = (bg- ch) (ch - of) (af- bg), 
Q = a 2 g 2 h 2 + b 2 h 2 f 2 + c 2 f 2 g 2 + a 2 6 2 c 2 , 
R = a 2 / 2 (a 2 +/ 2 ) + b 2 g 2 (6 2 + g 2 ) + c 2 h 2 (c 2 + h 2 ), 
we have 
•2P+Q-R = 0,