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