Page:Elementary Principles in Statistical Mechanics (1902).djvu/149

Rh Let us imagine an ensemble of systems distributed in phase according to the index of probability where $$\epsilon'$$ is any constant which is a possible value of the energy, except only the least value which is consistent with the values of the external coördinates, and $$c$$ and $$\omega$$ are other constants. We have therefore or  or again  From (404) we have  where $$\overline{A_1}|_\epsilon$$ denotes the average value of $$A_1$$ in those systems of the ensemble which have any same energy $$\epsilon$$. (This is the same thing as the average value of $$A_1$$ in a microcanonical ensemble of energy $$\epsilon$$.) The validity of the transformation is evident, if we consider separately the part of each integral which lies between two infinitesimally differing limits of energy. Integrating by parts, we get