Mathematical philosophy, a study of fate and freedom; lectures for educated laymen, by Cassius J. Keyser.

INVARIANCE 198
or real and unequal, or conjugate imaginary, numbers
according as b2-ac is equal to, or greater than, or less
than zero. Transforming (I) by replacing its x by x + Xy,
you will readily find that the transform of (I) is
(I') ax2+2(aX+b)xy+(aX2+2bX+c)y2, and that the
transform of (2) is the equation (2') ax2+2(aX+b)xy+
(a X2+2b X +c)y =0. The discriminant of the transform
(I') or (2') is (a X +b)2-a(aX2 +2bX+c), and this, you see,
reduces to b2 -ac exactly. You will notice that by allowing x to vary in value, we obtain an infinity of transformations-as x+2y, x+ey, x-~y, x+v/4y, and so on
-all of them similar in type, and a corresponding infinity
of transforms (I') or (2') of the same expression (I) or
equation (2); owing to the invariance of b2 -ac under each
and all of these transformations, we know that, if the
roots of (2) are equal, or real and unequal, or imaginary,
then the same is true of the roots of each one of the
infinitely many transform equations (2'). I have explained this simple fact,-small indeed as a mustard seed,
-so fully because, as already said, Lagrange's observation of it was the germ of the now vast and still growing
mathematical theory of invariance. Its early history
owns great names: Gauss who in I80o showed the discriminant of the ternary quadratic, ax2+by2+cz2+
2dxy +2exz+2fyz, to be invariant under the transformation replacing x by Ax+By+Cz, y by Dx+Ey+Fz and
z by Gx+Hy +Iz; Boole who in I841 established, among
other interesting results, the invariance of the discriminant of expressions involving an arbitrary number of
variables; Cayley who, incited by Boole's beautiful
results, assailed the problem of ascertaining all invariant
functions of the coefficients of an equation of degree n
and produced in rapid succession his great memoirs on