Search

Theorem ITI has both substantially stronger hypotheses and a stronger conclusion. The present paper obtains the stronger conclusion of Theorem II from the weaker hypotheses of Theorem I. For a more elaborate bibliography and historical comment on problems of the Sylvester type see [2], [3], and [4]. Precisely what the situation is when the requiremenit that one of the sets be infinite is dropped is not known. (See [2, Theorem 4.1 ].) This imatter is still under investigation.

Sylvester [7] proposed the following question in 1893. If a finite set of points in a plane is such that on the line determined by any two points of the set there is always a third point of the set, is the set collinear? Equivalently, given a finite planar set of non-collinear points, does there exist a line containing exactly two of the points?

The general solution is derived for a linear matrix differential equation of which the well-known equation $dX / dt = AX + XB$ is the simplest case. The method used relies on Kronecker matrix products, and some related results and extensions are also given.

The solution is effected by the method of Symmetric Functions.By a preliminary transformation * the equations of the original system are put into the form