10
ALGEBRAIC INVARIANTS
Hence the resultant of the new linear functions is
a' V
a b
a. /3
— A
a b
A' B'
A B
7 5
— LA
A B
and equals the product of the resultant r = aB—bA of the
given functions by A. Since this is true for every linear
homogeneous transformation of determinant A, we call r an
invariant of l and L of index unity, the factor which multiplies
r being here the first power of A.
Employing homogeneous coordinates for points on a line,
we see that l vanishes at the single point (b,—a) and that
L — 0 only at {B,—A). These two points are identical if
and only if b : a = B : A, i.e., if r = 0. The vanishing of the
invariant r thus indicates a geometrical property which is
independent of the choice of the points of reference used in
defining coordinates on the line; moreover, the property is
not changed by a projection of this line from an outside point
and a section by a new line. Thus r — 0 gives a projective
property.
Among the present transformations T are the very special
transformations given at the beginning of § 1. Of the four
functions there called invariants of l and L under those special
transformations, r alone is invariant under all of the present
transformations. Henceforth the term invariant will be used
only when the property of invariance holds for all linear homo
geneous transformations of the variables considered.
Our next example deals with the function
/=ax 2 +2bxy+cy 2 .
The transformation T (end of § 3) replaces / by
F=A £ 2 -\-2B£t?+Ct7 2 ,
in which
A = aa 2 + 2&cry+C7 2 ,
B = aa^-\-b{(x8-\-^y) A'CyS,
C = ap 2 +2b/38+c8 2 .
If the discriminant d = ac — b 2 of / is zero, / is the square
of a linear function of x and y, so that the transformed function