DYNAMICAL PRINCIPLES TO PHYSICAL PHENOMENA. 
493 
so that 
23 - d _ £ 
cl — 3 7] 
and 
23 -d £ 
3 £ + 7) 
again 
V8 = £+ v ; 
so that 
V (28 -d), 
rj = V(d- 8). 
Substituting these values for £ and rj in equation (29), we get 
K, lo S "T + A 1 - A s + R 1 = -'l- .(8°) 
This formula agrees substantially in form with one given by Professor Willard 
Gibbs in his paper on the “ Equilibrium of Heterogeneous Substances ” (‘ Transactions 
of the Connecticut Academy of Arts and Sciences,’ vol. 3, 1874-8, p. 239). In his 
paper on the vapour-densities of nitrogen tetroxide, formic acid, acetic acid, and per- 
chloride of phosphorus (‘American Journal of Science and Arts,’ vol. 18, 1879, p. 277), 
Professor Gibbs compares the density given by his formula with those found by 
various experimenters for the substances mentioned in the title of his paper, and he 
finds that the two sets of values agree very closely. 
Liquefaction and Solution. 
§ 9. We can apply the Hamiltonian principle to cases when a solid and liquid are in 
equilibrium in presence of each other, as, for example, when we have a mixture of ice 
and water, or a salt in a saturated solution of a liquid in which it can dissolve. 
Let us first consider the case of liquefaction and take the case of the melting of ice 
as the typical one. We must first find the value of L for a mixture of ice and water. 
The positional part of T for a solid or a liquid contains the term 
where v is the volume of the solid or liquid, v 0 is a constant, and dp/d6 is obtained on 
the supposition that v is constant. Thus the positional part of L for the mixture 
of ice and water equals 
