§66] 
CONCOMITANTS OF TERNARY FORMS 
93 
Under any ternary linear transformation 
T '• oct = £iXi -f- 7]iX2 + uX3 (i = 1, 2, 3) 
a x becomes a^Xi-{-(x v X2-\-a i X3, and/ becomes 
2^ i ^ t ATstX T \X S 2X l 3 = (oftAi d-aft^g-f-aj-Xs) • 
Thus a x behaves like a covariant of index zero of /. Also 
A TSt =oi{oiJa¡ t , 
a? a, a f 
ft ft ft = (afty )(£vt), 
7 i 7v 7f 
so that (a/37) behaves like an invariant of index unity of /. 
EXERCISES 
1. The discriminant of a ternary quadratic form a x 2 is | (a/37) 2 . 
2. The Jacobian of aft p x m , y x n is /»m (a/37)a x *- 1 / 8ft l-1 7ft~ 1 - 
3. The Hessian of a x n is the product of {a/3y) 2 a x n ~ 2: /3ft ~ 2 y x n ~ 2 by a 
constant. 
4. A ternary cubic form a x 3 =p x 3 = . . . has the invariants 
(a0y) (a/3 5) (ay 5) (@y 5), (a/3y) (a/3 5) (aye)(/3yct>)(5e<p) 2 . 
66. Concomitants of Ternary Forms. If u\, U2, U3 are 
constants, 
u x — u\X\ -\-U2X2 +U3X3 = 0 
represents a straight line in the point-coordinates xi, X2, X3. 
Since u\, U2, U3 determine this line, they are called its line- 
coordinates. If we give fixed values to xi, X2, X3 and let the 
line-coordinates ui, U2, U3 take all sets of values for which 
u x = 0, we obtain an infinite set of straight lines through the 
point (x\, X2, X3). Thus, for fixed x’s, u x = 0 is the equation 
of the point (xi, X2, X3) in line-coordinates. 
Under the linear transformation T, of § 65, whose deter 
minant (¿77ft is not zero, the line u x = 0 is replaced by 
Ux = U1X1 + U2X2 + U3X3 = 0, 
in which 333 
U\= 2 %iUt, U2— 2 rjiUi, U3— 2 %iUi. 
»=1 »=1 t=i