​

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 ${\displaystyle t'}$ fall between the limits ${\displaystyle p_{1}{}'}$ and ${\displaystyle p_{1}{}'+dp_{1}{}'}$, ${\displaystyle q_{1}{}'}$ and ${\displaystyle q_{1}{}'+dq_{1}{}'}$, etc., may be expressed by the formula

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

(48)

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

Let ${\displaystyle Q_{1}{}'}$, ${\displaystyle P_{1}{}'}$, etc. be the values of the coördinates and momenta which give the maximum value to ${\displaystyle \eta '}$, and let the general value of ${\displaystyle \eta '}$ be developed by Taylor's theorem according to ascending powers and products of the differences ${\displaystyle p_{1}{}'-P_{1}{}'}$, ${\displaystyle q_{1}{}'-Q_{1}{}'}$, 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

${\displaystyle \eta '=c-F',}$

(49)

where ${\displaystyle c}$ is independent of the differences ${\displaystyle p_{1}{}'-P_{1}{}'}$, ${\displaystyle q_{1}{}'-Q_{1}{}'}$, etc., and ${\displaystyle F'}$ is a homogeneous quadratic function of these ​differences. The terms of the first degree vanish in virtue of the maximum condition, which also requires that ${\displaystyle F'}$ must have a positive value except when all the differences mentioned vanish. If we set

${\displaystyle C=e^{c},}$

(50)

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

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

(51)

${\displaystyle C}$ 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 ${\displaystyle e^{\eta '}}$ or ${\displaystyle Ce^{-F'}}$ is practically zero except for very small values of the differences ${\displaystyle p_{1}{}'-P_{1}{}'}$, ${\displaystyle q_{1}{}'-Q_{1}{}'}$, etc. For very small values of these differences the approximation is evidently in general sufficient, for larger values of these differences the value of ${\displaystyle Ce^{-F'}}$ 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 (${\displaystyle t''}$) the maximum value of the coefficient of probability shall be the same as at the time ${\displaystyle t'}$ and that the phase ${\displaystyle (P_{1}{}'',Q_{1}{}'',\mathrm {etc.} )}$ which has this greatest probability-coefficient, shall be that which corresponds to the phase ${\displaystyle (P_{1}{}',Q_{1}{}',\mathrm {etc.} )}$, 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 ${\displaystyle t''}$ falls within the limits ${\displaystyle p_{1}{}''}$ and ${\displaystyle p_{1}{}''+dp_{1}{}''}$, ${\displaystyle q_{1}{}''}$ and ${\displaystyle q_{1}{}''+dq_{1}{}''}$, etc.,

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

(52)

​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 ​

${\displaystyle \left.{\begin{array}{c}p_{1}{}'-P_{1}{}'={\frac {dP_{1}{}'}{dP_{1}{}''}}(p_{1}{}''-P_{1}{}'')\ldots +{\frac {dP_{1}{}'}{dQ_{n}{}''}}(q_{n}{}''-Q_{n}{}'')\\\cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \\q_{n}{}'-Q_{n}{}'={\frac {dQ_{n}{}'}{dP_{1}{}''}}(p_{1}{}''-P_{1}{}'')\ldots +{\frac {dQ_{n}{}'}{dQ_{n}{}''}}(q_{n}{}''-Q_{n}{}'')\\\end{array}}\right\}}$

(55)

and as in ${\displaystyle F''}$ terms of higher degree than the second are to be neglected, these equations may be considered accurate for the purpose of the transformation required. Since by equation (33) the eliminant of these equations has the value unity, the discriminant of ${\displaystyle F''}$ will be equal to that of ${\displaystyle F'}$, as has already appeared from the consideration of the principle of conservation of probability of phase, which is, in fact, essentially the same as that expressed by equation (33).

At the time ${\displaystyle t'}$ the phases satisfying the equation

${\displaystyle F'=k,}$

(56)

where ${\displaystyle k}$ is any positive constant, have the probability-coefficient ${\displaystyle Ce^{-k}}$. At the time ${\displaystyle t''}$, the corresponding phases satisfy the equation

${\displaystyle F''=k,}$

(57)

and have the same probability-coefficient. So also the phases within the limits given by one or the other of these equations are corresponding phases, and have probability-coefficients greater than ${\displaystyle Ce^{-k}}$, while phases without these limits have less probability-coefficients. The probability that the phase at the time ${\displaystyle t'}$ falls within the limits ${\displaystyle F'=k}$ is the same as the probability that it falls within the limits ${\displaystyle F''=k}$ at the time ${\displaystyle t''}$, since either event necessitates the other. This probability may be evaluated as follows. We may omit the accents, as we need only consider a single time. Let us denote the extension-in-phase within the limits ${\displaystyle F=k}$ by ${\displaystyle U}$, and the probability that the phase falls within these limits by ${\displaystyle R}$, also the extension-in-phase within the limits ${\displaystyle F=1}$ by ${\displaystyle U_{1}}$. We have then by definition

${\displaystyle U=\mathop {\int \ldots \int } ^{F=k}dp_{1}\ldots dq_{n},}$

(58)

​

${\displaystyle R=\mathop {\int \ldots \int } ^{F=k}Ce^{-F}\,dp_{1}\ldots dq_{n},}$

(59)

${\displaystyle U_{1}=\mathop {\int \ldots \int } ^{F=1}dp_{1}\ldots dq_{n}.}$

(60)

But since ${\displaystyle F}$ is a homogeneous quadratic function of the differences

${\displaystyle p_{1}-P_{1},p_{2}-P_{2},\ldots q_{n}-Q_{n},}$

we have identically

${\displaystyle {\begin{aligned}&\mathop {\int \ldots \int } ^{F=k}d(p_{1}-P_{1})\ldots d(q_{n}-Q_{n})\\&=\mathop {\int \ldots \int } ^{kF=k}k^{n}\,d(p_{1}-P_{1})\ldots d(q_{n}-Q_{n})\\&=k^{n}\mathop {\int \ldots \int } ^{F=1}d(p_{1}-P_{1})\ldots d(q_{n}-Q_{n}).\end{aligned}}}$

That is

${\displaystyle U=k^{n}U_{1},}$

(61)

whence

${\displaystyle dU=U_{1}nk^{n-1}\,dk.}$

(62)

But if ${\displaystyle k}$ varies, equations (58) and (59) give

${\displaystyle dU=\mathop {\int \ldots \int } _{F=k}^{F=k+dk}dp_{1}\ldots dq_{n}}$

(63)

${\displaystyle dR=\mathop {\int \ldots \int } _{F=k}^{F=k+dk}Ce^{-F}\,dp_{1}\ldots dq_{n}}$

(64)

Since the factor ${\displaystyle Ce^{-F}}$ has the constant value ${\displaystyle Ce^{-k}}$ in the last multiple integral, we have

${\displaystyle dR=Ce^{-k}dU=CU_{1}ne^{-k}k^{n-1}\,dk,}$

(65)

${\displaystyle R=-CU_{1}n!e^{-k}\left(1+k+{\frac {k^{2}}{2}}+\ldots +{\frac {k^{n-1}}{(n-1)!}}\right)+\mathrm {const.} }$

(66)

We may determine the constant of integration by the condition that ${\displaystyle R}$ vanishes with ${\displaystyle k}$. This gives ​

${\displaystyle R=CU_{1}n!-CU_{1}n!e^{-k}\left(1+k+{\frac {k^{2}}{2}}+\ldots +{\frac {k^{n-1}}{(n-1)!}}\right).}$

(67)

We may determine the value of the constant ${\displaystyle U_{1}}$ by the condition that ${\displaystyle R=1}$ for ${\displaystyle k=\infty }$. This gives ${\displaystyle CU_{1}n!=1}$, and

${\displaystyle R=1-e^{-k}\left(1+k+{\frac {k^{2}}{2}}+\ldots +{\frac {k^{n-1}}{(n-1)!}}\right),}$

(68)

${\displaystyle U={\frac {k^{n}}{Cn!}}.}$

(69)

It is worthy of notice that the form of these equations depends only on the number of degrees of freedom of the system, being in other respects independent of its dynamical nature, except that the forces must be functions of the coördinates either alone or with the time.

If we write

${\displaystyle k_{R={\tfrac {1}{2}}}}$

for the value of ${\displaystyle k}$ which substituted in equation (68) will give ${\displaystyle R={\tfrac {1}{2}}}$, the phases determined by the equation

${\displaystyle F=k_{R={\tfrac {1}{2}}}}$

(70)

will have the following properties.

The probability that the phase falls within the limits formed by these phases is greater than the probability that it falls within any other limits enclosing an equal extension-in-phase. It is equal to the probability that the phase falls without the same limits.

These properties are analogous to those which in the theory of errors in the determination of a single quantity belong to values expressed by ${\displaystyle A\pm a}$, when ${\displaystyle A}$ is the most probable value, and ${\displaystyle a}$ the 'probable error.'