Introductory Remarks.—The principles set forth in this paper, though now (with the exception of the first theorem) published for the first time, were communicated to the French Academy of Sciences fifteen years ago, in a memoir entitled “ De l'Equilibre intérieur d'un Corps solide, élastique, et homogène,” and marked with the motto, “ Obvia conspicimus, nubem pellente Mathesi,” the receipt of which is acknowledged in the Comptes Rendus of the 6th April 1857.

(1.) Principle of Isorrhopic Axes.—The following theorem was first published in the “Philosophical Magazine” for December 1855.

Prop. I. “Theorem. Every self-balanced system of forces applied to a connected system of points is capable of resolution into three rectangular systems of parallel self-balanced forces applied to the same points.