4 
IY OF INVARIANTS. [567 
+ XY,f+X,g + Y,h + Z: 
R are 
! (c*Z 2 W 2 + h?X 2 F a ), 
567] AN IDENTICAL EQUATION CONNECTED WITH THE THEORY OF INVARIANTS. 55 
and, if in the first term we interchange 3 and 4, it becomes — (13.24) 2 (23.41), that 
is, + (14.23) (24.31) 2 , viz. it becomes equal to the second term. As regards Q the 
terms are all positive and become equal to each other; and the like as regards R: 
hence we have 
{12 V X V 2 V 3 V 4 (14.23) (24.31) 2 + 4 V 1 4 (23) 2 (34) 2 (42) 2 - 6 V X 2 V 4 2 (43) 4 (14) 2 } 
which, omitting a numerical factor 6.2.12 2 .2.24 2 .4, =3 5 .2 15 , is in fact the well-known 
equation 
il + JU~IH = 0, 
where 
U = (a, b, c, d, e) (x, yY, 
il = disct. (ax + by, bx + cy, cx + dy, dx + ey) (£, rj) 3 
= (ax + byf (dx + eyf + &c., 
I = ae— Ybd + 3c 2 , 
J = ace — ad 2 — b 2 c — c 3 -1- 2 bed, 
viz. attending only to the coefficient of ¿c 4 , this equation is 
a 2 d 2 + 4ac 3 + 4b 3 d — 36 2 c 2 — Qabcd + a (ace — ad 2 — b 2 e — c 3 + 2bed) + (ac — b 2 ) (ae — 4>bd + 3c 2 ) = 0. 
bove written equations of 
= 0. Hence taking U to 
ore U 1 = (a,...) (x 1 , y,Y, &c., 
each of them replaced by 
: negative terms, but that 
proper sign becomes equal 
one term thereof. Thus 
ms are — (a/) 2 bg + af(bg) 2 ,