About a year ago, while writing a Text-book on Heat in which the use of higher mathematics had to be avoided, it appeared tome that the phase-rule could be rigorously proved by a process which does not involve the deduction of the somewhat difficult thermodynamical equations used by Gibbs, Planck and others; Quite lately, however, I discovered that proofs somewhat similar to mine had been previously given by Nernst and Bancroft, and I must therefore not be understood to claim originality in thispaper. Seeing, however, that modern thermodynamics do not yet command in this country the interest they deserve, it will not be superfluous to draw the attention of the Society to the subject.