318 Proceedings of Royal Society of Edinburgh. [sess. 
the other two. A single substance, whose molecules are supposed 
to associate into groups of two or more, must still he looked upon 
as one substance from the point of view of the phase-rule. 
The condition of each phase is determined by (n+ 1) quantities, 
viz., the (n- 1) ratios in which the n substances occur in it and 
two additional quantities, say the temperature and the pressure. 
As, however, the last two are the same in all the phases, the total 
number of variables is (n- 1) r + 2. (If there are semi-permeable 
walls, the pressure is not the same in all the phases, and the 
phase-rule does not apply in its usual form.) 
In order to prove the phase-rule, we have to apply the second 
law of thermodynamics. For our purpose we may put it in this 
form, that the system must take up a condition of equilibrium ; 
otherwise we should get a perpetuum-mobile ; there must, there- 
fore, be an equation to be satisfied by the variables for every inde- 
pendent virtual reaction in the system. 
Apart from the conditions that the temperature and pressure 
are the same in all the phases which arise from the fact that an 
irreversible transference of heat or irreversible expansions are 
excluded, we thus obtain one equation for the virtual transition 
of every one of the n substances between every combination of 
two phases. If all these combinations had to be taken separately, 
r(r — 1 ) 
we should have \ x n e( l ua ^ ons ]n but; from the second 
law we conclude at once that the equilibrium between one phase 
and all the others separately involves that between every combina- 
tion of these last. The total number of equations is therefore 
(r -1) xn and the number of independent variables : 
(n—l)r + 2 - (r - l)n = n - r + 2. 
Q.E.D. 
