Page:Elementary Principles in Statistical Mechanics (1902).djvu/155

Rh and to that relating to the constant average energy, that

It is to be proved that Now in virtue of the first condition (424) we may cancel the constant term $$\psi/\Theta$$ in the parentheses in (426), and in virtue of the second condition (425) we may cancel the term $$\epsilon/\Theta$$. The proposition to be proved is thus reduced to which may be written, in virtue of the condition (424),  In this form its truth is evident for the same reasons which applied to (423).

Theorem III. If $$\Theta$$ is any positive constant, the average value in an ensemble of the expression $$\eta+\epsilon/\Theta$$ ($$\eta$$ denoting as usual the index of probability and $$\epsilon$$ the energy) is less when the ensemble is distributed canonically with modulus $$\Theta$$, than for any other distribution whatever.

In accordance with our usual notation let us write $$(\psi-\epsilon)/\Theta$$ for the index of the canonical distribution. In any other distribution let the index be $$(\psi-\epsilon)/\Theta + \Delta\eta$$.