94
ALGEBRAIC INVARIANTS
The equations obtained by solving these define a linear trans
formation T\ which expresses u\, uz, uz as linear functions
of U2, Uz and which is uniquely determined* by the
transformation T. Two sets of variables xi, xz, xz and u\, uz, uz,
transformed in this manner, are called contragredient.
A polynomial P(c, x, u) in the two sets of contragredient
variables and the coefficients c of certain forms fi(xi, xz, xz)
is called a mixed concomitant of index X of the /’s if, for every
linear transformation T of determinant A^O on xi, Xz, X3 and
the above defined transformation T\ on ui, uz, uz, the product
of P{c, x, u) by A x equals the same polynomial P{C, X, U)
in the new variables and coefficients C of the forms derived
from the /’s by the first transformation. For example, u x is
a concomitant of index zero of any set of forms.
In particular, if P does not involve the u’s, it is a covariant
(or invariant) of the f s. If it involves the u’s, but not the
x’s, it is called a contravariant of the/’s.
Since Ui=ui, Uz^Ur,, Uz = U{, we see by the last formula
in § 65, with 7 replaced by u, that (a/3w) behaves like a contra-
variant of index unity of a x n , and also like one of a x n , $ x m .
For the linear forms a x and p x , (a/3u) has an actual interpretation.
For f=a x 2 =p x 2 , where
/= O200#l 2 +fl020^2 2 +O002^3 2 T2(Zl01^1^3 4-2^011^2^3,
it may be shown that
¿200
Cno
Cm
Ux
C110
C020
Con
Ui
Cm
Coil
C002
U 3
Ztx
«2
U 3
0
By equating to zero this determinant (the bordered discriminant of
/), we obtain the line equation of the conic /=0.
67. Theorem. Every concomitant of a system of ternary forms
is a polynomial in u x and expressions of the types a x , (a@y), (a(3n).
* We have only to interchange the rows and columns in the matrix of T and
then take the inverse of the new matrix to obtain the matrix of the transforma
tion Ti. Similarly, x u x 2 are contragredient with u u u 2 , if we have T, § 40, and
«1 = (.V2 U1 — &U2) /ikv)) ( — ViV 1+ZiU2)/{iv)•
