Page:Elementary Principles in Statistical Mechanics (1902).djvu/46

where ${\displaystyle C}$ represents the same value as in the preceding formula, viz., the constant value of the maximum coefficient of probability, and ${\displaystyle F''}$ is a quadratic function of the differences ${\displaystyle p_{1}{}''-P_{1}{}''}$, ${\displaystyle q_{1}{}''-Q_{1}{}''}$, etc., the phase ${\displaystyle (P_{1}{}'',Q_{1}{}'',\mathrm {etc.} )}$ being that which at the time ${\displaystyle t''}$ corresponds to the phase ${\displaystyle (P_{1}{}',Q_{1}{}',\mathrm {etc.} )}$ at the time ${\displaystyle t'}$.

Now we have necessarily

${\displaystyle \int \ldots \int Ce^{-F'}\,dp_{1}{}'\ldots dq_{1}{}'=\int \ldots \int Ce^{-F''}\,dp_{1}{}''\ldots dq_{1}{}''=1,}$

(53)

when the integration is extended over all possible phases. It will be allowable to set ${\displaystyle \pm \infty }$ for the limits of all the coördinates and momenta, not because these values represent the actual limits of possible phases, but because the portions of the integrals lying outside of the limits of all possible phases will have sensibly the value zero. With ${\displaystyle \pm \infty }$ for limits, the equation gives

${\displaystyle {\frac {C\pi ^{n}}{\sqrt {f'}}}={\frac {C\pi ^{n}}{\sqrt {f''}}}=1,}$

(54)

where ${\displaystyle f'}$ is the discriminant^[1] of ${\displaystyle F'}$, and ${\displaystyle f''}$ that of ${\displaystyle F''}$. This discriminant is therefore constant in time, and like ${\displaystyle C}$ an absolute invariant in respect to the system of coördinates which may be employed. In dimensions, like ${\displaystyle C^{2}}$, it is the reciprocal of the 2nth power of the product of energy and time. Let us see precisely how the functions ${\displaystyle F'}$ and ${\displaystyle F''}$ are related. The principle of the conservation of the probability-coefficient requires that any values of the coördinates and momenta at the time ${\displaystyle t'}$ shall give the function ${\displaystyle F'}$ the same value as the corresponding coördinates and momenta at the time ${\displaystyle t''}$ give to ${\displaystyle F''}$. Therefore ${\displaystyle F''}$ may be derived from ${\displaystyle F'}$ by substituting for ${\displaystyle p_{1}{}',\ldots q_{1}{}'}$ their values in terms of ${\displaystyle p_{1}{}'',\ldots q_{1}{}''}$. Now we have approximately

1. ↑ This term is used to denote the determinant having for elements on the principal diagonal the coefficients of the squares in the quadratic function ${\displaystyle F'}$, and for its other elements the halves of the coefficients of the products in ${\displaystyle F'}$.