Page:Elementary Principles in Statistical Mechanics (1902).djvu/197

Rh with more than two degrees of freedom, the average values in the ensemble of $$d\phi/d\epsilon$$ for the two parts are equal to one another and to the value of same expression for the whole. In our usual notations if $$n_1 > 2$$, and $$n_2 > 2$$.

This analogy with temperature has the same incompleteness which was noticed with respect to $$d\epsilon/d\log V$$, viz., if two systems have such energies ($$\epsilon_1$$ and $$\epsilon_2$$) that and they are combined to form a third system with energy  we shall not have in general  Thus, if the energy is a quadratic function of the $$p$$'s and $$q$$'s, we have   where $$n_1$$, $$n_2$$, $$n_{12}$$, are the numbers of degrees of freedom of the separate and combined systems. But If the energy is a quadratic function of the $$p$$'s alone, the case would be the same except that we should have $$\tfrac 12 n_1$$, $$\tfrac 12 n_2$$, $$\tfrac 12 n_{12}$$, instead of $$n_1$$, $$n_2$$, $$n_{12}$$. In these particular cases, the analogy