Page:Elementary Principles in Statistical Mechanics (1902).djvu/135

where ${\displaystyle \psi '}$ denotes the value of ${\displaystyle \psi }$ for the modulus ${\displaystyle \Theta '}$. Since the last member of this formula vanishes for ${\displaystyle \epsilon =\infty }$, the less value represented by the first member must also vanish for the same value of ${\displaystyle \epsilon }$. Therefore the second member of (359), which differs only by a constant factor, vanishes at the upper limit. The case of the lower limit remains to be considered. Now

The second member of this formula evidently vanishes for the value of ${\displaystyle \epsilon }$, which gives ${\displaystyle V=0}$, whether this be finite or negative infinity. Therefore, the second member of (359) vanishes at the lower limit also, and we have or This equation, which is subject to no restriction in regard to the value of ${\displaystyle n}$, suggests a connection or analogy between the function of the energy of a system which is represented by ${\displaystyle e^{-\phi }V}$ and the notion of temperature in thermodynamics. We shall return to this subject in Chapter XIV.

If ${\displaystyle n>2}$, the second member of (359) may easily be shown to vanish for any of the following values of ${\displaystyle u}$ viz.: ${\displaystyle \phi }$, ${\displaystyle e^{\phi }}$, ${\displaystyle \epsilon }$, ${\displaystyle \epsilon ^{m}}$, where ${\displaystyle m}$ denotes any positive number. It will also vanish, when ${\displaystyle n>4}$, for ${\displaystyle u=d\phi /d\epsilon }$, and when ${\displaystyle n>2h}$ for ${\displaystyle u=e^{-\phi }\,d^{h}V/d\epsilon ^{h}}$. When the second member of (359) vanishes, and ${\displaystyle n>2}$, we may write

We thus obtain the following equations:

If ${\displaystyle n>2}$,