Most of the theorems in question occur at the outset of a paper by Professor Cayley, entitled, “Chapters on the Analytical Geometry of n Dimensions” they constitute, in fact, Chapter I. The first theorem I should prefer, for the present, to enunciate as follows:-

If m determinants of the nthorder all have the same n − 1 columns in common, and all vanish, then every determinant of the nthorder whose n columns are chosen from the m + n − 1 different columns must vanish likewise.