494 
PROFESSOR J. J. THOMSON ON SOME APPLICATIONS OF 
where V is the mean potential energy of the mixture. By Hamilton's principle 
this expression must be stationary when there is equilibrium, the temperature remain¬ 
ing constant. Let us suppose unit mass of ice to melt: the change in the first terms is 
where is the volume of unit mass of water, and v x the volume of unit mass of ice. 
The increase in the potential energy is equal to the latent heat under the pressure p 
of water, so that Hamilton’s principle gives 
Now Planck has shown* that 
where P, the pressure to which both the ice and water are subjected, is regarded as a 
function of the temperature alone. 
Hence we have, by equation (32), 
-<) = *,.(34) 
the well-known equation connecting the change in the melting point of ice with 
the change in the pressure to which it is subjected. In this equation X is the increase 
in the potential energy when unit mass of the ice melts. It will depend to some 
extent upon external circumstances ; thus, if the water is of such a shape that the 
area of its free surface changes when ice melts, then, on account of the energy due to 
surface-tension, X will depend upon the change in the surface. As the volume 
diminishes as the ice melts, the surface will in general diminish, so that the energy 
due to this cause will be diminished by the liquefaction, and the effect of pressure 
upon the freezing-point increased. Since the volume changes, work is done by or 
against the external pressure : thus X will be a function of the pressure. If X 0 be the 
value of X when the pressure is zero, X the value when the pressure is p, we may 
easily prove that 
k = —p (h - <)> 
so that X diminishes as the pressure increases, and the effect on the freezing-point 
of a given increment of pressure will increase as the pressure increases. 
* ‘Wiedemann’s Annalen,’ vol. 13, p. 541. 
