Page:Elementary Principles in Statistical Mechanics (1902).djvu/44

CHAPTER II.

APPLICATION OF THE PRINCIPLE OF CONSERVATION OF EXTENSION-IN-PHASE TO THE THEORY OF ERRORS.

Let us now proceed to combine the principle which has been demonstrated in the preceding chapter and which in its different applications and regarded from different points of view has been variously designated as the conservation of density-in-phase, or of extension-in-phase, or of probability of phase, with those approximate relations which are generally used in the 'theory of errors.'

We suppose that the differential equations of the motion of a system are exactly known, but that the constants of the integral equations are only approximately determined. It is evident that the probability that the momenta and coördinates at the time $t'$ fall between the limits $p_{1}{}'$ and $p_{1}{}'+dp_{1}{}'$, $q_{1}{}'$ and $q_{1}{}'+dq_{1}{}'$, etc., may be expressed by the formula

$e^{\eta '}\,dp_{1}{}'\ldots dq_{1}{}',$

(48)

where $\eta '$ (the index of probability for the phase in question) is a function of the coördinates and momenta and of the time.

Let $Q_{1}{}'$, $P_{1}{}'$, etc. be the values of the coördinates and momenta which give the maximum value to $\eta '$, and let the general value of $\eta '$ be developed by Taylor's theorem according to ascending powers and products of the differences $p_{1}{}'-P_{1}{}'$, $q_{1}{}'-Q_{1}{}'$, etc. and let us suppose that we have a sufficient approximation without going beyond terms of the second degree in these differences. We may therefore set

$\eta '=c-F',$

(49)

where $c$ is independent of the differences $p_{1}{}'-P_{1}{}'$, $q_{1}{}'-Q_{1}{}'$, etc., and $F'$ is a homogeneous quadratic function of these