What is an invariant?
In collaboration with Cayley, he developed the theory of `forms’ (or `quantics’ as called by Cayley), through which they came to be known as invariant twins. Sylvester published, between 1854 and 1878, a dozen papers on forms _ homogeneous polynomials and their invariants. The most important cases in analytic geometry and physics are the quadratic forms in two and three variables. When equated to a constant, these represent conics and quadrics. For example, Ax2 + 2Bxy + Cy2 when equated to a non-zero constant represents an ellipse, a parabola or a hyperbola according to B2- AC less than, equal to, or greater than zero. Now if the form is transformed under a rotation of axes about the origin into the new form Ax2 + 2Bxy+ Cy2 then (B)2 – AC = B2 – AC the expression B2 – AC is an invariant under such a transformation. Neither Lagrange nor Gauss seemed to have noticed the above simple but remarkable algebraical phenomenon which turned out to be the germ of a vast theory of algebraic invari
