The important researches of Joule on the thermal circumstances connected with the expansion and compression of air, and the admirable reasoning upon them expressed in his paper, “On the Changes of Temperature produced by the Rarefaction and Condensation of Air;” especially the way in which he takes into account any mechanical effect that may be externally produced, or internally lost in fluid friction, have introduced an entirely new method of treating questions regarding the physical properties of fluids. The object of the present paper is to show how, by the use of this new method, in connection with the principles explained in the author's preceding paper on the Dynamical Theory of Heat, a complete theoretical view may be obtained of the phenomena experimented on by Joule, and to point out some of the objects to be attained by a continuation and extension of his experimental researches.