§68] 
QUATERNARY FORMS 
97 
68. Quaternary Forms. For a x =a\Xi-\-. . .+«4X4, 
n on n ? n 
— Oix —Px —Tz — ®x. 
has the determinant (aftyb) of order 4 as a symbolic invariant 
of index unity. Any invariant of / can be expressed as a 
polynomial in such determinan tal factors; any co variant as 
a polynomial in them and factors of type a x . In the equation 
u x = 0 of a plane, ui, . . ., u± are called plane-coordinates. 
The mixed concomitants defined as in § 66 are expressible 
in terms of u x and factors like a x , {a(37 5), (aj3yu). For geometrical 
reasons, we extend that definition of mixed concomitants to 
polynomials P{c, x, u, v), where Vi,. . ., 24 as well as m,. . ., W4 
are contragredient to xi, . . ., #4. There may now occur 
the additional type of factor 
(pifiuv) =(ai^2-Oi2^l)(usV4:— U4Vs)+. . • + («3/34 -«4/3s) {U1V2—U2V1). 
These six combinations of the #’s and v’s are called the line- 
coordinates of the intersection of the planes u x = 0, v x = 0. For 
instance, {a^uv) 2 = 0 is the condition that this line of inter 
section shall touch the quadric surface a x 2 = 0. 
We have not considered concomitants involving also a 
third set of variables wi, . . ., W4, contragredient with the x’s. 
For, in 
U\X\ + . . .+W4#4 = 0, • . -\~V4X4 = 0, 
WlXl+. . .+it»4X4 = 0, 
xi, . . ., X4 are proportional to the three-rowed determinants 
of the matrix of coefficients, so that {auvw) is essentially a x .