Page:Elementary Principles in Statistical Mechanics (1902).djvu/145

which shows that $d\phi \,d\log V$ approaches the value unity when $n$ is very great.

If a system consists of two parts, having separate energies, we may obtain equations similar in form to the preceding, which relate to the system as thus divided.^[1] We shall distinguish quantities relating to the parts by letters with suffixes, the same letters without suffixes relating to the whole system. The extension-in-phase of the whole system within any given limits of the energies may be represented by the double integral

$\iint dV_{1}\,dV_{2}$

taken within those limits, as appears at once from the definitions of Chapter VIII. In an ensemble distributed with uniform density within those limits, and zero density outside, the average value of any function of $\epsilon _{1}$ and $\epsilon _{2}$ is given by the quotient

${\frac {\iint u\,dV_{1}\,dV_{2}}{\iint dV_{1}\,dV_{2}}}$

which may also be written^[2]

${\frac {\iint u\,e^{\phi _{1}}\,d\epsilon \,dV_{2}}{\iint e^{\phi _{1}}\,d\epsilon \,dV_{2}}}$

If we make the limits of integration $\epsilon$ and $\epsilon +d\epsilon$, we get the

1. If this condition is rigorously fulfilled, the parts will have no influence on each other, and the ensemble formed by distributing the whole microcanonically is too arbitrary a conception to have a real interest. The principal interest of the equations which we shall obtain will be in cases in which the condition is approximately fulfilled. But for the purposes of a theoretical discussion, it is of course convenient to make such a condition absolute. Compare Chapter IV, pp. 35 ff., where a similar condition is considered in connection with canonical ensembles.
2. Where the analytical transformations are identical in form with those on the preceding pages, it does not appear necessary to give all the steps with the same detail.