Page:Elementary Principles in Statistical Mechanics (1902).djvu/122

corresponding to other than very small values of ${\displaystyle \epsilon _{q}-\epsilon _{q}'}$ may be regarded as a vanishing quantity.

This gives

${\displaystyle e^{\phi }=e^{\phi _{p}'+\phi _{q}'}\left({\frac {-2\pi }{{\Big (}{\frac {d^{2}\phi _{p}}{d\epsilon _{p}^{2}}}{\Big )}'+{\Big (}{\frac {d^{2}\phi _{q}}{d\epsilon _{q}^{2}}}{\Big )}'}}\right)^{\frac {1}{2}}}$

(313)

or

${\displaystyle \phi =\phi _{p}'+\phi _{q}'+{\tfrac {1}{2}}\log(2\pi )-{\tfrac {1}{2}}\log {\bigg [}-{\Big (}{\frac {d^{2}\phi _{p}}{d\epsilon _{p}^{2}}}{\Big )}'-{\Big (}{\frac {d^{2}\phi _{q}}{d\epsilon _{q}^{2}}}{\Big )}'{\bigg ]}.}$

(314)

From this equation, with (289), (300) and (309), we may determine the value of ${\displaystyle \phi }$ corresponding to any given value of ${\displaystyle \epsilon }$, when ${\displaystyle \phi _{q}}$ is known as function of ${\displaystyle \epsilon _{q}}$.

Any two systems may be regarded as together forming a third system. If we have ${\displaystyle V}$ or ${\displaystyle \phi }$ given as function of ${\displaystyle \epsilon }$ for any two systems, we may express by quadratures ${\displaystyle V}$ and ${\displaystyle \phi }$ for the system formed by combining the two. If we distinguish by the suffixes ${\displaystyle (~)_{1}}$, ${\displaystyle (~)_{2}}$, ${\displaystyle (~)_{12}}$ the quantities relating to the three systems, we have easily from the definitions of these quantities

${\displaystyle V_{12}=\int \int dV_{1}\,dV_{2}=\int V_{2}\,dV_{1}=\int V_{1}\,dV_{2}=\int V_{1}e^{\phi _{2}}\,d\epsilon _{2},}$

(315)

${\displaystyle e^{\phi _{12}}=\int e^{\phi _{2}}\,dV_{1}=\int e^{\phi _{1}}\,dV_{2}=\int e^{\phi _{1}+\phi _{2}}\,d\epsilon _{2},}$

(316)

where the double integral is to be taken within the limits

${\displaystyle V_{1}=0,V_{2}=0,\mathrm {and} \epsilon _{1}+\epsilon _{2}=\epsilon _{12},}$

and the variables in the single integrals are connected by the last of these equations, while the limits are given by the first two, which characterize the least possible values of ${\displaystyle \epsilon _{1}}$ and ${\displaystyle \epsilon _{2}}$ respectively.

It will be observed that these equations are identical in form with those by which ${\displaystyle V}$ and ${\displaystyle \phi }$ are derived from ${\displaystyle V_{p}}$ or ${\displaystyle \phi _{p}}$ and ${\displaystyle V_{q}}$ or ${\displaystyle \phi _{q}}$, except that they do not admit in the general case those transformations which result from substituting for ${\displaystyle V_{p}}$ or ${\displaystyle \phi _{p}}$ the particular functions which these symbols always represent.