1. After having established this remarkable theorem in the manner shown in the text, the author attempted to prove it by direct integration, but only succeeded in doing so upwards of a year later, when he obtained the demonstration published in a paper Propositions in the theory of attraction Camb. Math. Journ. Nov. 1842 which appeared almost contemporaneously with a paper by M. Sturm in Liouville's Journal, containing the same demonstration; exactly the same demonstration, as the author afterwards (in 1845) found, had been given fourteen years earlier by Green.