The equation of a rational plane curve derived from its parametric equations (Second paper)
Bulletin of the American Mathematical Society
a point which lies in the polar r -2 flat of a qua f (a) = 0. Whence it follows that any r -2 flat Sw^a* = 0 is invariant to each of the infinitesimal transformations of Y represented by the poles of this flat qua f (a) = 0; and thus, if the poles of this flat do not all lie in an r -2 flat, it follows that the infinitesimal transformations of G represented by points in the r -2 flat Huiai = 0 generate an invariant subgroup of G. If the adjoint group V has a quadratic invariant of non-zero
... nt of non-zero discriminant, for a proper choice of the X's, we shall have Cijk + Cikj = 0 (i, j, k = 1, 2, • • -, r). In this case the condition that f (a) shall be invariant to V is that df(a) df(a) \ where Ei is the matrix whose constituent in the juth row and pth column is d VfJL (i, \x, v = 1, 2, • • -, r); or, what is the same thing, is that for all values of the a's. Whence it follows, if f (a) is a second invariant of T, that the infinitesimal transformation *LuiXi is commutative with every infinitesimal transformation of G represented by a pole, qua f (a) = 0, of the r -2 flat S^a t = 0; and, if these poles do not all lie in any r -2 flat, it follows that XuiXi is an exceptional infinitesimal transformation.