Page:Elementary Principles in Statistical Mechanics (1902).djvu/45

differences. The terms of the first degree vanish in virtue of the maximum condition, which also requires that $F'$ must have a positive value except when all the differences mentioned vanish. If we set

$C=e^{c},$

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we may write for the probability that the phase lies within the limits considered

$Ce^{-F'}\,dp_{1}{}'\ldots dq_{1}{}'.$

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$C$ is evidently the maximum value of the coefficient of probability at the time considered.

In regard to the degree of approximation represented by these formulæ, it is to be observed that we suppose, as is usual in the 'theory of errors' that the determination (explicit or implicit) of the constants of motion is of such precision that the coefficient of probability $e^{\eta '}$ or $Ce^{-F'}$ is practically zero except for very small values of the differences $p_{1}{}'-P_{1}{}'$, $q_{1}{}'-Q_{1}{}'$, etc. For very small values of these differences the approximation is evidently in general sufficient, for larger values of these differences the value of $Ce^{-F'}$ will be sensibly zero, as it should be, and in this sense the formula will represent the facts.

We shall suppose that the forces to which the system is subject are functions of the coördinates either alone or with the time. The principle of conservation of probability of phase will therefore apply, which requires that at any other time ($t''$) the maximum value of the coefficient of probability shall be the same as at the time $t'$ and that the phase $(P_{1}{}'',Q_{1}{}'',\mathrm {etc.} )$ which has this greatest probability-coefficient, shall be that which corresponds to the phase $(P_{1}{}',Q_{1}{}',\mathrm {etc.} )$, i. e., which is calculated from the same values of the constants of the integral equations of motion.

We may therefore write for the probability that the phase at the time $t''$ falls within the limits $p_{1}{}''$ and $p_{1}{}''+dp_{1}{}''$, $q_{1}{}''$ and $q_{1}{}''+dq_{1}{}''$, etc.,

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

(52)