This paper commences with a condensed re-statement of the fundamental principles and formulæ of the Dynamical Theory of Heat, from the first six parts of the author's treatment of the subject previously communicated to the Royal Society of Edinburgh, and his articles "On the Thermo-elastic Properties of Matter," in the "Quarterly Mathematical Journal" (April 1855), and on "Thermo-magnetism," and "Thermo-electricity," in Nichol's Cyclopedia (Edinburgh 1860).