The discovery of the configuration known as a Determinantal System of Points was made by Dr W. P. Milne while investigating the problem of the “Generation of a cubic curve by apolar pencils of lines.” In the Proc. L. M. S., Ser. 2, Vol. 15, Part 4, he obtained a complete solution, but in unsymmetrical form. He therefore suggested to me that as he was up to that time unable to find a symmetrical solution for the general cubic I should investigate Determinantal Systems for the case of rational curves. The results of my investigations are given in this and the ensuing paper, and have enabled Dr Milne to solve the general problem in symmetrical form in a paper which will appear in the Proc. L. M. S. at an early date. I commence from the result given in the paper by Dr Milne, entitled Determinantal Systems of Points, in the Proc. E. M. S., Vol XXXIV., Part 2.