Elementary Principles in Statistical Mechanics/Chapter XV

by Josiah Willard Gibbs

CHAPTER XV.

SYSTEMS COMPOSED OF MOLECULES.

The nature of material bodies is such, that especial interest attaches to the dynamics of systems composed of a great number of entirely similar particles, or, it may be, of a great number of particles of several kinds, all of each kind being entirely similar to each other. We shall therefore proceed to consider systems composed of such particles, whether in great numbers or otherwise, and especially to consider the statistical equilibrium of ensembles of such systems. One of the variations to be considered in regard to such systems is a variation in the numbers of the particles of the various kinds which it contains, and the question of statistical equilibrium between two ensembles of such systems relates in part to the tendencies of the various kinds of particles to pass from the one to the other.

First of all, we must define precisely what is meant by statistical equilibrium of such an ensemble of systems. The essence of statistical equilibrium is the permanence of the number of systems which fall within any given limits with respect to phase. We have therefore to define how the term "phase" is to be understood in such cases. If two phases differ only in that certain entirely similar particles have changed places with one another, are they to be regarded as identical or different phases? If the particles are regarded as indistinguishable, it seems in accordance with the spirit of the statistical method to regard the phases as identical. In fact, it might be urged that in such an ensemble of systems as we are considering no identity is possible between the particles of different systems except that of qualities, and if $\nu$ particles of one system are described as entirely similar to one another and to $\nu$ of another system, nothing remains on which to base the indentification of any particular particle of the first system with any particular particle of the second. And this would be true, if the ensemble of systems had a simultaneous objective existence. But it hardly applies to the creations of the imagination. In the cases which we have been considering, and in those which we shall consider, it is not only possible to conceive of the motion of an ensemble of similar systems simply as possible cases of the motion of a single system, but it is actually in large measure for the sake of representing more clearly the possible cases of the motion of a single system that we use the conception of an ensemble of systems. The perfect similarity of several particles of a system will not in the least interfere with the identification of a particular particle in one case with a particular particle in another. The question is one to be decided in accordance with the requirements of practical convenience in the discussion of the problems with which we are engaged.

Our present purpose will often require us to use the terms phase, density-in-phase, statistical equilibrium, and other connected terms on the supposition that phases are not altered by the exchange of places between similar particles. Some of the most important questions with which we are concerned have reference to phases thus defined. We shall call them phases determined by generic definitions, or briefly, generic phases. But we shall also be obliged to discuss phases defined by the narrower definition (so that exchange of position between similar particles is regarded as changing the phase), which will be called phases determined by specific definitions, or briefly, specific phases. For the analytical description of a specific phase is more simple than that of a generic phase. And it is a more simple matter to make a multiple integral extend over all possible specific phases than to make one extend without repetition over all possible generic phases.

It is evident that if $\nu _{1},\nu _{2}\ldots \nu _{h}$, are the numbers of the different kinds of molecules in any system, the number of specific phases embraced in one generic phase is represented by the continued product ${\nu _{1}}!\,{\nu _{2}}!\ldots {\nu _{h}}!$ and the coefficient of probability of a generic phase is the sum of the probability-coefficients of the specific phases which it represents. When these are equal among themselves, the probability-coefficient of the generic phase is equal to that of the specific phase multiplied by ${\nu _{1}}!\,{\nu _{2}}!\ldots {\nu _{h}}!$. It is also evident that statistical equilibrium may subsist with respect to generic phases without statistical equilibrium with respect to specific phases, but not vice versa.

Similar questions arise where one particle is capable of several equivalent positions. Does the change from one of these positions to another change the phase? It would be most natural and logical to make it affect the specific phase, but not the generic. The number of specific phases contained in a generic phase would then be ${\nu _{1}}!\,\kappa _{1}^{\nu _{1}}\ldots {\nu _{h}}!\,\kappa _{h}^{\nu _{h}}$, where $\kappa _{1}\ldots \kappa _{h}$ denote the numbers of equivalent positions belonging to the several kinds of particles. The case in which a $\kappa$ is infinite would then require especial attention. It does not appear that the resulting complications in the formulae would be compensated by any real advantage. The reason of this is that in problems of real interest equivalent positions of a particle will always be equally probable. In this respect, equivalent positions of the same particle are entirely unlike the $\nu !$ different ways in which $\nu$ particles may be distributed in $\nu$ different positions. Let it therefore be understood that in spite of the physical equivalence of different positions of the same particle they are to be considered as constituting a difference of generic phase as well as of specific. The number of specific phases contained in a generic phase is therefore always given by the product ${\nu _{1}}!\,{\nu _{2}}!\ldots {\nu _{h}}!$.

Instead of considering, as in the preceding chapters, ensembles of systems differing only in phase, we shall now suppose that the systems constituting an ensemble are composed of particles of various kinds, and that they differ not only in phase but also in the numbers of these particles which they contain. The external coördinates of all the systems in the ensemble are supposed, as heretofore, to have the same value, and when they vary, to vary together. For distinction, we may call such an ensemble a grand ensemble, and one in which the systems differ only in phase a petit ensemble. A grand ensemble is therefore composed of a multitude of petit ensembles. The ensembles which we have hitherto discussed are petit ensembles.

Let $\nu _{1},\ldots \nu _{h}$, etc., denote the numbers of the different kinds of particles in a system, $\epsilon$ its energy, and $q_{1},\ldots q_{n}$, $p_{1},\ldots p_{n}$ its coördinates and momenta. If the particles are of the nature of material points, the number of coördinates ($n$) of the system will be equal to $3\nu _{1}\ldots +3\nu _{h}$. But if the particles are less simple in their nature, if they are to be treated as rigid solids, the orientation of which must be regarded, or if they consist each of several atoms, so as to have more than three degrees of freedom, the number of coördinates of the system will be equal to the sum of $\nu _{1}$, $\nu _{2}$, etc., multiplied each by the number of degrees of freedom of the kind of particle to which it relates.

Let us consider an ensemble in which the number of systems having $\nu _{1},\ldots \nu _{h}$ particles of the several kinds, and having values of their coördinates and momenta lying between the limits $q_{1}$ and $q_{1}+dq_{1}$, $p_{1}$ and $p_{1}+dp_{1}$, etc., is represented by the expression

${\frac {Ne^{\frac {\Omega +\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}-\epsilon }{\Theta }}}{{\nu _{1}}!\ldots {\nu _{h}}!}}\,dp_{1}\ldots dq_{n},$

(4998)

where $N$, $\Omega$, $\Theta$, $\mu _{1},\ldots \mu _{h}$ are constants, $N$ denoting the total number of systems in the ensemble. The expression

${\frac {Ne^{\frac {\Omega +\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}-\epsilon }{\Theta }}}{{\nu _{1}}!\ldots {\nu _{h}}!}}$

(499)

evidently represents the density-in-phase of the ensemble within the limits described, that is, for a phase specifically defined. The expression

${\frac {e^{\frac {\Omega +\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}-\epsilon }{\Theta }}}{{\nu _{1}}!\ldots {\nu _{h}}!}}$

(500)

is therefore the probability-coefficient for a phase specifically defined. This has evidently the same value for all the ${\nu _{1}}!\ldots {\nu _{h}}!$ phases obtained by interchanging the phases of particles of the same kind. The probability-coefficient for a generic phase will be ${\nu _{1}}!\ldots {\nu _{h}}!$ times as great, viz.,

$e^{\frac {\Omega +\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}-\epsilon }{\Theta }}.$

(501)

We shall say that such an ensemble as has been described is canonically distributed, and shall call the constant $\Theta$ its modulus. It is evidently what we have called a grand ensemble. The petit ensembles of which it is composed are canonically distributed, according to the definitions of Chapter IV, since the expression

${\frac {e^{\frac {\Omega +\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}}{\Theta }}}{{\nu _{1}}!\ldots {\nu _{h}}!}}$

(502)

is constant for each petit ensemble. The grand ensemble, therefore, is in statistical equilibrium with respect to specific phases.

If an ensemble, whether grand or petit, is identical so far as generic phases are concerned with one canonically distributed, we shall say that its distribution is canonical with respect to generic phases. Such an ensemble is evidently in statistical equilibrium with respect to generic phases, although it may not be so with respect to specific phases.

If we write $\mathrm {H}$ for the index of probability of a generic phase in a grand ensemble, we have for the case of canonical distribution

$\mathrm {H} ={\frac {\Omega +\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}-\epsilon }{\Theta }}$

(503)

It will be observed that the $\mathrm {H}$ is a linear function of $\epsilon$ and $\nu _{1},\ldots \nu _{h}$; also that whenever the index of probability of generic phases in a grand ensemble is a linear function of $\epsilon$, $\nu _{1},\ldots \nu _{h}$, the ensemble is canonically distributed with respect to generic phases.

The constant $\Omega$ we may regard as determined by the equation

$N=\Sigma _{\nu _{1}}\ldots \Sigma _{\nu _{h}}\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}{\frac {Ne^{\frac {\Omega +\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}-\epsilon }{\Theta }}}{{\nu _{1}}!\ldots {\nu _{h}}!}}\,dp_{1}\ldots dq_{n},$

(504)

or

$e^{-{\frac {\Omega }{\Theta }}}=\Sigma _{\nu _{1}}\ldots \Sigma _{\nu _{h}}{\frac {e^{\frac {\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}}{\Theta }}}{{\nu _{1}}!\ldots {\nu _{h}}!}}\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}e^{-{\frac {\epsilon }{\Theta }}}\,dp_{1}\ldots dq_{n},$

(505)

where the multiple sum indicated by $\Sigma _{\nu _{1}}\ldots \Sigma _{\nu _{h}}$ includes all terms obtained by giving to each of the symbols $\nu _{1}\ldots \nu _{h}$ all integral values from zero upward, and the multiple integral (which is to be evaluated separately for each term of the multiple sum) is to be extended over all the (specific) phases of the system having the specified numbers of particles of the various kinds. The multiple integral in the last equation is what we have represented by $e^{-{\frac {\psi }{\Theta }}}$. See equation (92). We may therefore write

$e^{-{\frac {\Omega }{\Theta }}}=\Sigma _{\nu _{1}}\ldots \Sigma _{\nu _{h}}{\frac {e^{\frac {\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}-\psi }{\Theta }}}{{\nu _{1}}!\ldots {\nu _{h}}!}}.$

(506)

It should be observed that the summation includes a term in which all the symbols $\nu _{1}\ldots \nu _{h}$ have the value zero. We must therefore recognize in a certain sense a system consisting of no particles, which, although a barren subject of study in itself, cannot well be excluded as a particular case of a system of a variable number of particles. In this case $\epsilon$ is constant, and there are no integrations to be performed. We have therefore^[1]

$e^{-{\frac {\psi }{\Theta }}}=e^{-{\frac {\epsilon }{\Theta }}},\quad {\mathit {i.\,e.}},\quad \psi =\epsilon .$

The value of $\epsilon _{p}$ is of course zero in this case. But the value of $\epsilon _{q}$ contains an arbitrary constant, which is generally determined by considerations of convenience, so that $\epsilon _{q}$ and $\epsilon$ do not necessarily vanish with $\nu _{1},\ldots \nu _{h}$.

Unless $-\Omega$ has a finite value, our formulae become illusory. We have already, in considering petit ensembles canonically distributed, found it necessary to exclude cases in which $-\psi$ has not a finite value.^[2] The same exclusion would here make $-\psi$ finite for any finite values of $\nu _{1}\ldots \nu _{h}$. This does not necessarily make a multiple series of the form (506) finite. We may observe, however, that if for all values of $\nu _{1}\ldots \nu _{h}$

$-\psi \leq c_{0}+c_{1}\nu _{1},\ldots +c_{h}\nu _{h},$

(507)

where $c_{0},c_{1},\ldots c_{h}$ are constants or functions of $\Theta$,

$e^{-{\frac {\Omega }{\Theta }}}\leq \Sigma _{\nu _{1}}\ldots \Sigma _{\nu _{h}}{\frac {e^{\frac {c_{0}+(\mu _{1}+c_{1})\nu _{1}\ldots +(\mu _{h}+c_{h})\nu _{h}}{\Theta }}}{{\nu _{1}}!\ldots {\nu _{h}}!}}$

i. e.,

$e^{-{\frac {\Omega }{\Theta }}}\leq e^{\frac {c_{0}}{\Theta }}\Sigma _{\nu _{1}}{\frac {e^{{\frac {\mu _{1}+c_{1}}{\Theta }}\nu _{1}}}{{\nu _{1}}!}}\ldots \Sigma _{\nu _{h}}{\frac {e^{{\frac {\mu _{h}+c_{h}}{\Theta }}\nu _{h}}}{{\nu _{h}}!}}$

i. e.,

$e^{-{\frac {\Omega }{\Theta }}}\leq e^{\frac {c_{0}}{\Theta }}e^{e^{\frac {\mu _{1}+c_{1}}{\Theta }}}\ldots e^{e^{\frac {\mu _{h}+c_{h}}{\Theta }}}$

i. e.,

$-{\frac {\Omega }{\Theta }}\leq {\frac {c_{0}}{\Theta }}+e^{\frac {\mu _{1}+c_{1}}{\Theta }}\ldots +e^{\frac {\mu _{h}+c_{h}}{\Theta }}$

(508)

The value of $-\Omega$ will therefore be finite, when the condition (507) is satisfied. If therefore we assume that $-\Omega$ is finite, we do not appear to exclude any cases which are analogous to those of nature.^[3]

The interest of the ensemble which has been described lies in the fact that it may be in statistical equilbrium, both in respect to exchange of energy and exchange of particles, with other grand ensembles canonically distributed and having the same values of $\Theta$ and of the coefficients $\mu _{1}$, $\mu _{2}$, etc., when the circumstances are such that exchange of energy and of particles are possible, and when equilibrium would not subsist, were it not for equal values of these constants in the two ensembles.

With respect to the exchange of energy, the case is exactly the same as that of the petit ensembles considered in Chapter IV, and needs no especial discussion. The question of exchange of particles is to a certain extent analogous, and may be treated in a somewhat similar manner. Let us suppose that we have two grand ensembles canonically distributed with respect to specific phases, with the same value of the modulus and of the coefficients $\mu _{1}\ldots \mu _{h}$, and let us consider the ensemble of all the systems obtained by combining each system of the first ensemble with each of the second.

The probability-coefficient of a generic phase in the first ensemble may be expressed by

$e^{\frac {\Omega '+\mu _{1}\nu _{1}'\ldots +\mu _{h}\nu _{h}'-\epsilon '}{\Theta }}$

(509)

The probability-coefficient of a specific phase will then be expressed by

${\frac {e^{\frac {\Omega '+\mu _{1}\nu _{1}'\ldots +\mu _{h}\nu _{h}'-\epsilon '}{\Theta }}}{{\nu _{1}'}!\ldots {\nu _{h}'}!}},$

(510)

since each generic phase comprises ${\nu _{1}}!\ldots {\nu _{h}}!$ specific phases. In the second ensemble the probability-coefficients of the generic and specific phases will be

$e^{\frac {\Omega ''+\mu _{1}\nu _{1}''\ldots +\mu _{h}\nu _{h}''-\epsilon ''}{\Theta }},$

(511)

and

${\frac {e^{\frac {\Omega ''+\mu _{1}\nu _{1}''\ldots +\mu _{h}\nu _{h}''-\epsilon ''}{\Theta }}}{{\nu _{1}''}!\ldots {\nu _{h}''}!}}.$

(512)

The probability-coefficient of a generic phase in the third ensemble, which consists of systems obtained by regarding each system of the first ensemble combined with each of the second as forming a system, will be the product of the probability-coefficients of the generic phases of the systems combined, and will therefore be represented by the formula

$e^{\frac {\Omega '''+\mu _{1}\nu _{1}'''\ldots +\mu _{h}\nu _{h}'''-\epsilon '''}{\Theta }},$

(513)

where $\Omega '''=\Omega '+\Omega '$, $\epsilon '''=\epsilon '+\epsilon ''$, $\nu _{1}'''=\nu _{1}'+\nu _{1}''$, etc. It will be observed that $\nu _{1}'''$, etc., represent the numbers of particles of the various kinds in the third ensemble, and $\epsilon '''$ its energy; also that $\Omega '''$ is a constant. The third ensemble is therefore canonically distributed with respect to generic phases.

If all the systems in the same generic phase in the third ensemble were equably distributed among the ${\nu _{1}'''}!\ldots {\nu _{h}'''}!$ specific phases which are comprised in the generic phase, the probability-coefficient of a specific phase would be

${\frac {e^{\frac {\Omega '''+\mu _{1}\nu _{1}'''\ldots +\mu _{h}\nu _{h}'''-\epsilon '''}{\Theta }}}{{\nu _{1}'''}!\ldots {\nu _{h}'''}!}}..$

(514)

In fact, however, the probability-coefficient of any specific phase which occurs in the third ensemble is

${\frac {e^{\frac {\Omega '''+\mu _{1}\nu _{1}'''\ldots +\mu _{h}\nu _{h}'''-\epsilon '''}{\Theta }}}{{\nu _{1}'}!\ldots {\nu _{h}'}!\,{\nu _{1}''}!\ldots {\nu _{h}''}!}}.,$

(515)

which we get by multiplying the probability-coefficients of specific phases in the first and second ensembles. The difference between the formulae (514) and (515) is due to the fact that the generic phases to which (513) relates include not only the specific phases occurring in the third ensemble and having the probability-coefficient (515), but also all the specific phases obtained from these by interchange of similar particles between two combined systems. Of these the probability-coefficient is evidently zero, as they do not occur in the ensemble.

Now this third ensemble is in statistical equilibrium, with respect both to specific and generic phases, since the ensembles from which it is formed are so. This statistical equilibrium is not dependent on the equality of the modulus and the co-efficients $\mu _{1},\ldots \mu _{h}$ in the first and second ensembles. It depends only on the fact that the two original ensembles were separately in statistical equilibrium, and that there is no interaction between them, the combining of the two ensembles to form a third being purely nominal, and involving no physical connection. This independence of the systems, determined physically by forces which prevent particles from passing from one system to the other, or coming within range of each other's action, is represented mathematically by infinite values of the energy for particles in a space dividing the systems. Such a space may be called a diaphragm.

If we now suppose that, when we combine the systems of the two original ensembles, the forces are so modified that the energy is no longer infinite for particles in all the space forming the diaphragm, but is diminished in a part of this space, so that it is possible for particles to pass from one system to the other, this will involve a change in the function $\epsilon '''$ which represents the energy of the combined systems, and the equation $\epsilon '''=\epsilon '+\epsilon ''$ will no longer hold. Now if the co-efficient of probability in the third ensemble were represented by (513) with this new function $\epsilon '''$, we should have statistical equilibrium, with respect to generic phases, although not to specific. But this need involve only a trifling change in the distribution of the third ensemble,^[4] a change represented by the addition of comparatively few systems in which the transference of particles is taking place to the immense number obtained by combining the two original ensembles. The difference between the ensemble which would be in statistical equilibrium, and that obtained by combining the two original ensembles may be diminished without limit, while it is still possible for particles to pass from one system to another. In this sense we may say that the ensemble formed by combining the two given ensembles may still be regarded as in a state of (approximate) statistical equilibrium with respect to generic phases, when it has been made possible for particles to pass between the systems combined, and when statistical equilibrium for specific phases has therefore entirely ceased to exist, and when the equilibrium for generic phases would also have entirely ceased to exist, if the given ensembles had not been canonically distributed, with respect to generic phases, with the same values of $\Theta$ and $\mu _{1},\ldots \mu _{h}$.

It is evident also that considerations of this kind will apply separately to the several kinds of particles. We may diminish the energy in the space forming the diaphragm for one kind of particle and not for another. This is the mathematical expression for a "semipermeable" diaphragm. The condition necessary for statistical equilibrium where the diaphragm is permeable only to particles to which the suffix $(~)_{1}$ relates will be fulfilled when $\mu _{1}$ and $\Theta$ have the same values in the two ensembles, although the other coefficients $\mu _{2}$, etc., may be different.

This important property of grand ensembles with canonical distribution will supply the motive for a more particular examination of the nature of such ensembles, and especially of the comparative numbers of systems in the several petit ensembles which make up a grand ensemble, and of the average values in the grand ensemble of some of the most important quantities, and of the average squares of the deviations from these average values.

The probability that a system taken at random from a grand ensemble canonically distributed will have exactly $\nu _{1},\ldots \nu _{h}$ particles of the various kinds is expressed by the multiple integral

$\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}{\frac {e^{\frac {\Omega +\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}-\epsilon }{\Theta }}}{{\nu _{1}}!\ldots {\nu _{h}}!}}\,dp_{1}\ldots dq_{n},$

(516)

or

${\frac {e^{\frac {\Omega +\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}-\psi }{\Theta }}}{{\nu _{1}}!\ldots {\nu _{h}}!}}.$

(517)

This may be called the probability of the petit ensemble $(\nu _{1},\ldots \nu _{h})$. The sum of all such probabilities is evidently unity. That is,

$\Sigma _{\nu _{1}}\ldots \Sigma _{\nu _{h}}{\frac {e^{\frac {\Omega +\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}-\psi }{\Theta }}}{{\nu _{1}}!\ldots {\nu _{h}}!}}=1,$

(518)

which agrees with (506).

The average value in the grand ensemble of any quantity $u$, is given by the formula

${\overline {u}}=\Sigma _{\nu _{1}}\ldots \Sigma _{\nu _{h}}\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}{\frac {ue^{\frac {\Omega +\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}-\epsilon }{\Theta }}}{{\nu _{1}}!\ldots {\nu _{h}}!}}\,dp_{1}\ldots dq_{n}.$

(519)

If $u$ is a function of $\nu _{1},\ldots \nu _{h}$ alone, i. e., if it has the same value in all systems of any same petit ensemble, the formula reduces to

${\overline {u}}=\Sigma _{\nu _{1}}\ldots \Sigma _{\nu _{h}}{\frac {ue^{\frac {\Omega +\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}-\psi }{\Theta }}}{{\nu _{1}}!\ldots {\nu _{h}}!}}.$

(520)

Again, if we write ${\overline {u}}|_{\rm {grand}}$ and ${\overline {u}}|_{\rm {petit}}$ to distinguish averages in the grand and petit ensembles, we shall have

${\overline {u}}|_{\rm {grand}}=\Sigma _{\nu _{1}}\ldots \Sigma _{\nu _{h}}{\overline {u}}|_{\rm {petit}}{\frac {e^{\frac {\Omega +\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}-\psi }{\Theta }}}{{\nu _{1}}!\ldots {\nu _{h}}!}}.$

(521)

In this chapter, in which we are treating of grand ensembles, ${\overline {u}}$ will always denote the average for a grand ensemble. In the preceding chapters, ${\overline {u}}$ has always denoted the average for a petit ensemble.

Equation (505), which we repeat in a slightly different form, viz.,

$e^{-{\frac {\Omega }{\Theta }}}=\Sigma _{\nu _{1}}\ldots \Sigma _{\nu _{h}}\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}{\frac {e^{\frac {\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}-\epsilon }{\Theta }}}{{\nu _{1}}!\ldots {\nu _{h}}!}}\,dp_{1}\ldots dq_{n},,$

(522)

shows that $\Omega$ is a function of $\Theta$ and $\mu _{1},\ldots \mu _{h}$; also of the external coördinates $a_{1}$, $a_{2}$, etc., which are involved implicitly in $\epsilon$. If we differentiate the equation regarding all these quantities as variable, we have

${\begin{aligned}&e^{-{\frac {\Omega }{\Theta }}}{\bigg (}-{\frac {d\Omega }{\Theta }}+{\frac {\Omega }{\Theta ^{2}}}d\Theta {\bigg )}=\\&\quad {}-{\frac {d\Theta }{\Theta ^{2}}}\Sigma _{\nu _{1}}\ldots \Sigma _{\nu _{h}}\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}{\frac {(\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}-\epsilon )e^{\frac {\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}-\epsilon }{\Theta }}}{{\nu _{1}}!\ldots {\nu _{h}}!}}\,dp_{1}\ldots dq_{n}\\&\quad {}+{\frac {d\mu _{1}}{\Theta }}\Sigma _{\nu _{1}}\ldots \Sigma _{\nu _{h}}\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}{\frac {\nu _{1}e^{\frac {\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}-\epsilon }{\Theta }}}{{\nu _{1}}!\ldots {\nu _{h}}!}}\,dp_{1}\ldots dq_{n}\\&\quad {}+\mathrm {etc.} \\&\quad {}-{\frac {da_{1}}{\Theta }}\Sigma _{\nu _{1}}\ldots \Sigma _{\nu _{h}}\mathop {\int \ldots \int } _{\rm {phases}}^{\rm {all}}{\frac {d\epsilon }{da_{1}}}\,{\frac {e^{\frac {\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}-\epsilon }{\Theta }}}{{\nu _{1}}!\ldots {\nu _{h}}!}}\,dp_{1}\ldots dq_{n}\\&\quad {}-\mathrm {etc.} \\\end{aligned}}$

(523)

If we multiply this equation by $e^{\frac {\Omega }{\Theta }}$, and set as usual $A_{1}$, $A_{2}$, etc., for $-d\epsilon /da_{1}$, $-d\epsilon /da_{2}$, etc., we get in virtue of the law expressed by equation (519),

${\begin{aligned}-{\frac {d\Omega }{\Theta }}+{\frac {\Omega }{\Theta ^{2}}}d\Theta &=-{\frac {d\Theta }{\Theta ^{2}}}(\mu _{1}{\overline {\nu }}_{1}\ldots \mu _{h}{\overline {\nu }}_{h}-{\overline {\epsilon }})\\&\quad {}+{\frac {d\mu _{1}}{\Theta }}{\overline {\nu }}_{1}+{\frac {d\mu _{2}}{\Theta }}{\overline {\nu }}_{2}+\mathrm {etc.} \\&\quad {}+{\frac {da_{1}}{\Theta }}{\overline {A}}_{1}+{\frac {da_{2}}{\Theta }}{\overline {A}}_{2}+\mathrm {etc.} ;\end{aligned}}$

(524)

that is,

$d\Omega ={\frac {\Omega +\mu _{1}{\overline {\nu }}_{1}\ldots \mu _{h}{\overline {\nu }}_{h}-{\overline {\epsilon }}}{\Theta }}\,d\Theta -\Sigma {\overline {\nu }}_{1}\,d\mu _{1}-\Sigma {\overline {A}}_{1}\,da_{1}.$

(525)

Since equation (503) gives

${\frac {\Omega +\mu _{1}{\overline {\nu }}_{1}\ldots +\mu _{h}{\overline {\nu }}_{h}-{\overline {\epsilon }}}{\Theta }}={\overline {\mathrm {H} }},$

(526)

the preceding equation may be written

$d\Omega ={\overline {\mathrm {H} }}\,d\Theta -\Sigma {\overline {\nu }}_{1}\,d\mu _{1}-\Sigma {\overline {A}}_{1}\,da_{1}.$

(527)

Again, equation (526) gives

$d\Omega +\Sigma \mu _{1}\,d{\overline {\nu }}_{1}+\Sigma {\overline {\nu }}_{1}\,d\mu _{1}-d{\overline {\epsilon }}=\Theta \,d{\overline {\mathrm {H} }}+{\overline {\mathrm {H} }}\,d\Theta .$

(528)

Eliminating $d\Omega$ from these equations, we get

$d{\overline {\epsilon }}=-\Theta \,d{\overline {\mathrm {H} }}+\Sigma \mu _{1}\,d{\overline {\nu }}_{1}-\Sigma {\overline {A}}_{1}\,da_{1}.$

(529)

If we set

$\Psi ={\overline {\epsilon }}+\Theta {\overline {\mathrm {H} }},$

(530)

$d\Psi =d\epsilon +\Theta \,d{\overline {\mathrm {H} }}+{\overline {\mathrm {H} }}\,d\Theta ,$

(531)

we have

$d\Psi ={\overline {\mathrm {H} }}\,d\Theta +\Sigma \mu _{1}\,d{\overline {\nu }}_{1}-\Sigma {\overline {A}}_{1}\,da_{1}.$

(532)

The corresponding thermodynamic equations are

$d\epsilon =T\,d\eta +\Sigma \mu _{1}\,dm_{1}-\Sigma A_{1}\,da_{1},$

(533)

$\psi =\epsilon -T\eta ,$

(534)

$d\psi =-\eta \,dT+\Sigma \mu _{1}\,dm_{1}-\Sigma A_{1}\,da_{1}.$

(535)

These are derived from the thermodynamic equations (114) and (117) by the addition of the terms necessary to take account of variation in the quantities ($m_{1}$, $m_{2}$, etc.) of the several substances of which a body is composed. The correspondence of the equations is most perfect when the component substances are measured in such units that $m_{1}$, $m_{2}$, etc., are proportional to the numbers of the different kinds of molecules or atoms. The quantities $\mu _{1}$, $\mu _{2}$, etc., in these thermodynamic equations may be defined as differential coefficients by either of the equations in which they occur.^[5]

If we compare the statistical equations (529) and (532) with (114) and (112), which are given in Chapter IV, and discussed in Chapter XIV, as analogues of thermodynamic equations, we find considerable difference. Beside the terms corresponding to the additional terms in the thermodynamic equations of this chapter, and beside the fact that the averages are taken in a grand ensemble in one case and in a petit in the other, the analogues of entropy, $\mathrm {H}$ and $\eta$, are quite different in definition and value. We shall return to this point after we have determined the order of magnitude of the usual anomalies of $\nu _{1},\ldots \nu _{h}$.

If we differentiate equation (518) with respect to $\mu _{1}$, and multiply by $\Theta$, we get

$\Sigma _{\nu _{1}}\ldots \Sigma _{\nu _{h}}{\bigg (}{\frac {d\Omega }{d\mu _{1}}}+\nu _{1}{\bigg )}{\frac {e^{\frac {\Omega +\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}-\psi }{\Theta }}}{{\nu _{1}}!\ldots {\nu _{h}}!}}=0,$

(536)

whence $d\Omega /d\mu _{1}=-{\overline {\nu }}_{1}$, which agrees with (527). Differentiating again with respect to $\mu _{1}$, and to $\mu _{2}$, and setting

${\frac {d\Omega }{d\mu _{1}}}=-{\overline {\nu }}_{1},\quad {\frac {d\Omega }{d\mu _{2}}}=-{\overline {\nu }}_{2},$

we get

$\Sigma _{\nu _{1}}\ldots \Sigma _{\nu _{h}}{\bigg (}{\frac {d^{2}\Omega }{d\mu _{1}^{2}}}+{\frac {(\nu _{1}-{\overline {\nu }}_{1})^{2}}{\Theta }}{\bigg )}{\frac {e^{\frac {\Omega +\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}-\psi }{\Theta }}}{{\nu _{1}}!\ldots {\nu _{h}}!}}=0,$

(537)

$\Sigma _{\nu _{1}}\ldots \Sigma _{\nu _{h}}{\bigg (}{\frac {d^{2}\Omega }{d\mu _{1}\,d\mu _{2}}}+{\frac {(\nu _{1}-{\overline {\nu }}_{1})(\nu _{2}-{\overline {\nu }}_{2})}{\Theta }}{\bigg )}{\frac {e^{\frac {\Omega +\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}-\psi }{\Theta }}}{{\nu _{1}}!\ldots {\nu _{h}}!}}=0.$

(538)

The first members of these equations represent the average values of the quantities in the principal parentheses. We have therefore

${\overline {(\nu _{1}-{\overline {\nu }}_{1})^{2}}}={\overline {\nu _{1}^{2}}}-{\overline {\nu }}_{1}^{2}=-\Theta {\frac {d^{2}\Omega }{d\mu _{1}^{2}}}=\Theta {\frac {d{\overline {\nu }}_{1}}{d\mu _{1}}},$

(539)

${\overline {(\nu _{1}-{\overline {\nu }}_{1})(\nu _{2}-{\overline {\nu }}_{2})}}={\overline {\nu _{1}\nu _{2}}}-{\overline {\nu }}_{1}{\overline {\nu }}_{2}=-\Theta {\frac {d^{2}\Omega }{d\mu _{1}\,d\mu _{2}}}=\Theta {\frac {d{\overline {\nu }}_{1}}{d\mu _{2}}}=\Theta {\frac {d{\overline {\nu }}_{2}}{d\mu _{1}}}.$

(540)

From equation (539) we may get an idea of the order of magnitude of the divergences of $\nu _{1}$ from its average value in the ensemble, when that average value is great. The equation may be written

${\frac {(\nu _{1}-{\overline {\nu }}_{1})^{2}}{{\overline {\nu }}_{1}^{2}}}={\frac {\Theta }{{\overline {\nu }}_{1}^{2}}}{\frac {d{\overline {\nu }}_{1}}{d\mu _{1}}},$

(541)

The second member of this equation will in general be small when ${\overline {\nu }}_{1}$ is great. Large values are not necessarily excluded, but they must be confined within very small limits with respect to $\mu$. For if

${\frac {\overline {(\nu _{1}-{\overline {\nu }}_{1})^{2}}}{{\overline {\nu }}_{1}^{2}}}>{\frac {2}{{\overline {\nu }}_{1}^{\frac {1}{2}}}},$

(542)

for all values of $\mu _{1}$ between the limits $\mu _{1}'$ and $\mu _{1}''$, we shall have between the same limits

${\frac {\Theta }{{\overline {\nu }}_{1}^{\frac {3}{2}}}}\,d{\overline {\nu }}_{1}>d\mu _{1},$

(543)

and therefore

${\tfrac {1}{2}}\Theta {\bigg (}{\frac {1}{{\overline {\nu }}_{1}'^{\frac {1}{2}}}}-{\frac {1}{{\overline {\nu }}_{1}''^{\frac {1}{2}}}}{\bigg )}>\mu _{1}''-\mu _{1}'.$

(544)

The difference $\mu _{1}''-\mu _{1}'$ is therefore numerically a very small quantity. To form an idea of the importance of such a difference, we should observe that in formula (498) $\mu _{1}$ is multiplied by $\nu _{1}$ and the product subtracted from the energy. A very small difference in the value of $\mu _{1}$ may therefore be important. But since $\nu \Theta$ is always less than the kinetic energy of the system, our formula shows that $\mu _{1}''-\mu _{1}'$, even when multiplied by ${\overline {\nu }}_{1}'$ or ${\overline {\nu }}_{1}''$, may still be regarded as an insensible quantity.

We can now perceive the leading characteristics with respect to properties sensible to human faculties of such an ensemble as we are considering (a grand ensemble canonically distributed), when the average numbers of particles of the various kinds are of the same order of magnitude as the number of molecules in the bodies which are the subject of physical experiment. Although the ensemble contains systems having the widest possible variations in respect to the numbers of the particles which they contain, these variations are practically contained within such narrow limits as to be insensible, except for particular values of the constants of the ensemble. This exception corresponds precisely to the case of nature, when certain thermodynamic quantities corresponding to $\Theta$, $\mu _{1}$, $\mu _{2}$, etc., which in general determine the separate densities of various components of a body, have certain values which make these densities indeterminate, in other words, when the conditions are such as determine coexistent phases of matter. Except in the case of these particular values, the grand ensemble would not differ to human faculties of perception from a petit ensemble, viz., any one of the petit ensembles which it contains in which ${\overline {\nu }}_{1}$, ${\overline {\nu }}_{2}$, etc., do not sensibly differ from their average values.

Let us now compare the quantities $\mathrm {H}$ and $\eta$, the average values of which (in a grand and a petit ensemble respectively) we have seen to correspond to entropy. Since

$\mathrm {H} ={\frac {\Omega +\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}-\epsilon }{\Theta }},$

and

$\eta ={\frac {\psi -\epsilon }{\Theta }},$

$\mathrm {H} -\eta ={\frac {\Omega +\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}-\psi }{\Theta }}.$

(545)

A part of this difference is due to the fact that $\mathrm {H}$ relates to generic phases and $\eta$ to specific. If we write $\eta _{\rm {gen}}$ for the index of probability for generic phases in a petit ensemble, we have

$\eta _{\rm {gen}}=\eta +\log {\nu _{1}}!\ldots {\nu _{h}}!,$

(546)

$H-\eta =\mathrm {H} -\eta _{\rm {gen}}+\log {\nu _{1}}!\ldots {\nu _{h}}!,$

(547)

$\mathrm {H} -\eta _{\rm {gen}}={\frac {\Omega +\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}-\psi }{\Theta }}-\log {\nu _{1}}!\ldots {\nu _{h}}!.$

(548)

This is the logarithm of the probability of the petit ensemble ($\nu _{1}\ldots \nu _{h}$).^[6] If we set

${\frac {\psi _{\rm {gen}}-\epsilon }{\Theta }}=\eta _{\rm {gen}},$

(549)

which corresponds to the equation

${\frac {\psi -\epsilon }{\Theta }}=\eta ,$

(550)

we have

$\psi _{\rm {gen}}=\psi +\Theta \log {\nu _{1}}!\ldots {\nu _{h}}!,$

and

$\mathrm {H} -\eta _{\rm {gen}}={\frac {\Omega +\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}-\psi _{\rm {gen}}}{\Theta }}.$

(551)

This will have a maximum when^[7]

${\frac {d\psi _{\rm {gen}}}{d\nu _{1}}}=\mu _{1},\qquad {\frac {d\psi _{\rm {gen}}}{d\nu _{2}}}=\mu _{2},\qquad \mathrm {etc.}$

(552)

Distinguishing values corresponding to this maximum by accents, we have approximately, when $\nu _{1},\ldots \nu _{h}$ are of the same order of magnitude as the numbers of molecules in ordinary bodies,

${\begin{aligned}\mathrm {H} -\eta _{\rm {gen}}&={\frac {\Omega +\mu _{1}\nu _{1}\ldots +\mu _{h}\nu _{h}-\psi _{\rm {gen}}}{\Theta }}\\&={\frac {\Omega +\mu _{1}\nu _{1}'\ldots +\mu _{h}\nu _{h}'-\psi _{\rm {gen}}'}{\Theta }}\\&\!\!\!\!\!\!\!\!\!\!\!\!\!\!{}-{\bigg (}{\frac {d^{2}\psi _{\rm {gen}}}{d\nu _{1}^{2}}}{\bigg )}'{\frac {(\Delta \nu _{1})^{2}}{2\Theta }}-{\bigg (}{\frac {d^{2}\psi _{\rm {gen}}}{d\nu _{1}\,d\nu _{2}}}{\bigg )}'{\frac {\Delta \nu _{1}\Delta \nu _{2}}{\Theta }}\ldots -{\bigg (}{\frac {d^{2}\psi _{\rm {gen}}}{d\nu _{h}^{2}}}{\bigg )}'{\frac {(\Delta \nu _{h})^{2}}{2\Theta }}\end{aligned}},$

(553)

$e^{\mathrm {H} -\eta _{\rm {gen}}}=e^{C}e^{-{\big (}{\frac {d^{2}\psi _{\rm {gen}}}{d\nu _{1}^{2}}}{\big )}'{\frac {(\Delta \nu _{1})^{2}}{2\Theta }}-{\big (}{\frac {d^{2}\psi _{\rm {gen}}}{d\nu _{1}\,d\nu _{2}}}{\big )}'{\frac {\Delta \nu _{1}\Delta \nu _{2}}{\Theta }}\ldots -{\big (}{\frac {d^{2}\psi _{\rm {gen}}}{d\nu _{h}^{2}}}{\big )}'{\frac {(\Delta \nu _{h})^{2}}{2\Theta }}},$

(554)

where

$C={\frac {\Omega +\mu _{1}\nu _{1}'\ldots +\mu _{h}\nu _{h}'-\psi _{\rm {gen}}'}{\Theta }},$

(555)

and

$\Delta \nu _{1}=\nu _{1}-\nu _{1}',\quad \Delta \nu _{2}=\nu _{2}-\nu _{2}',\quad \mathrm {etc.}$

(556)

This is the probability of the system ($\nu _{1}\ldots \nu _{h}$). The probabilty that the values of $\nu _{1},\ldots \nu _{h}$ lie within given limits is given by the multiple integral

$\int \ldots \int e^{C}e^{-{\big (}{\frac {d^{2}\psi _{\rm {gen}}}{d\nu _{1}^{2}}}{\big )}'{\frac {(\Delta \nu _{1})^{2}}{2\Theta }}-{\big (}{\frac {d^{2}\psi _{\rm {gen}}}{d\nu _{1}\,d\nu _{2}}}{\big )}'{\frac {\Delta \nu _{1}\Delta \nu _{2}}{\Theta }}\ldots -{\big (}{\frac {d^{2}\psi _{\rm {gen}}}{d\nu _{h}^{2}}}{\big )}'{\frac {(\Delta \nu _{h})^{2}}{2\Theta }}}\,d\nu _{1}\ldots d\nu _{h}.$

(557)

This shows that the distribution of the grand ensemble with respect to the values of $\nu _{1},\ldots \nu _{h}$ follows the "law of errors" when $\nu _{1}',\ldots \nu _{h}'$ are very great. The value of this integral for the limits $\pm \infty$ should be unity. This gives

$e^{C}{\frac {(2\pi \Theta )^{\frac {h}{2}}}{D^{\frac {h}{2}}}}=1,$

(558)

or

$C={\tfrac {1}{2}}\log D-{\frac {h}{2}}\log(2\pi \Theta ),$

(559)

where

$D={\begin{vmatrix}{\Big (}{\frac {d^{2}\psi _{\rm {gen}}}{d\nu _{1}^{2}}}{\Big )}'&{\Big (}{\frac {d^{2}\psi _{\rm {gen}}}{d\nu _{1}\,d\nu _{2}}}{\Big )}'&\ldots &{\Big (}{\frac {d^{2}\psi _{\rm {gen}}}{d\nu _{1}\,d\nu _{h}}}{\Big )}'\\{\Big (}{\frac {d^{2}\psi _{\rm {gen}}}{d\nu _{2}\,d\nu _{1}}}{\Big )}'&{\Big (}{\frac {d^{2}\psi _{\rm {gen}}}{d\nu _{2}^{2}}}{\Big )}'&\ldots &{\Big (}{\frac {d^{2}\psi _{\rm {gen}}}{d\nu _{2}\,d\nu _{h}}}{\Big )}'\\\ldots &\ldots &\ldots &\ldots \\\ldots &\ldots &\ldots &\ldots \\{\Big (}{\frac {d^{2}\psi _{\rm {gen}}}{d\nu _{h}\,d\nu _{1}}}{\Big )}'&{\Big (}{\frac {d^{2}\psi _{\rm {gen}}}{d\nu _{h}\,d\nu _{2}}}{\Big )}'&\ldots &{\Big (}{\frac {d^{2}\psi _{\rm {gen}}}{d\nu _{h}^{2}}}{\Big )}'\end{vmatrix}}$

(560)

that is,

$D={\begin{vmatrix}{\big (}{\frac {d\mu _{1}}{d\nu _{1}}}{\big )}'&{\big (}{\frac {d\mu _{1}}{d\nu _{2}}}{\big )}'&\ldots &{\big (}{\frac {d\mu _{1}}{d\nu _{h}}}{\big )}'&\\{\big (}{\frac {d\mu _{2}}{d\nu _{1}}}{\big )}'&{\big (}{\frac {d\mu _{2}}{d\nu _{2}}}{\big )}'&\ldots &{\big (}{\frac {d\mu _{2}}{d\nu _{h}}}{\big )}'&\\\ldots &\ldots &\ldots &\ldots \\\ldots &\ldots &\ldots &\ldots \\{\big (}{\frac {d\mu _{h}}{d\nu _{1}}}{\big )}'&{\big (}{\frac {d\mu _{h}}{d\nu _{2}}}{\big )}'&\ldots &{\big (}{\frac {d\mu _{h}}{d\nu _{h}}}{\big )}'&\end{vmatrix}}$

(561)

Now, by (553), we have for the first approximation

${\overline {\mathrm {H} }}-{\overline {\eta }}_{\rm {gen}}=C={\tfrac {1}{2}}\log D-{\frac {h}{2}}\log(2\pi \Theta ),$

(562)

and if we divide by the constant $K$,^[8] to reduce these quantities to the usual unit of entropy,

${\frac {{\overline {\mathrm {H} }}-{\overline {\eta }}_{\rm {gen}}}{K}}={\frac {\log D-h\log(2\pi \Theta )}{2K}}$

(563)

This is evidently a negligible quantity, since $K$ is of the same order of magnitude as the number of molecules in ordinary bodies. It is to be observed that ${\overline {\eta }}_{\rm {gen}}$ is here the average in the grand ensemble, whereas the quantity which we wish to compare with ${\overline {\mathrm {H} }}$ is the average in a petit ensemble. But as we have seen that in the case considered the grand ensemble would appear to human observation as a petit ensemble, this distinction may be neglected.

The differences therefore, in the case considered, between the quantities which may be represented by the notations^[9]

${\overline {\mathrm {H} _{\rm {gen}}}}|_{\rm {grand}},\quad {\overline {\eta _{\rm {gen}}}}|_{\rm {grand}},\quad {\overline {\eta _{\rm {gen}}}}|_{\rm {petit}}$

are not sensible to human faculties. The difference

${\overline {\eta _{\rm {gen}}}}|_{\rm {petit}}-{\overline {\eta _{\rm {spec}}}}|_{\rm {petit}}={\nu _{1}}!\ldots {\nu _{h}}!,$

and is therefore constant, so long as the numbers $\nu _{1},\ldots \nu _{h}$ are constant. For constant values of these numbers, therefore, it is immaterial whether we use the average of $\eta _{\rm {gen}}$ or of $\eta$ for entropy, since this only affects the arbitrary constant of integration which is added to entropy. But when the numbers $\nu _{1},\ldots \nu _{h}$ are varied, it is no longer possible to use the index for specific phases. For the principle that the entropy of any body has an arbitrary additive constant is subject to limitation, when different quantities of the same substance are concerned. In this case, the constant being determined for one quantity of a substance, is thereby determined for all quantities of the same substance.

To fix our ideas, let us suppose that we have two identical fluid masses in contiguous chambers. The entropy of the whole is equal to the sum of the entropies of the parts, and double that of one part. Suppose a valve is now opened, making a communication between the chambers. We do not regard this as making any change in the entropy, although the masses of gas or liquid diffuse into one another, and al-though the same process of diffusion would increase the entropy, if the masses of fluid were different. It is evident, therefore, that it is equilibrium with respect to generic phases, and not with respect to specific, with which we have to do in the evaluation of entropy, and therefore, that we must use the average of $\mathrm {H}$ or of $\eta _{\rm {gen}}$, and not that of $\eta$, as the equivalent of entropy, except in the thermodynamics of bodies in which the number of molecules of the various kinds is constant.

---

1. This conclusion may appear a little strained. The original definition of $\psi$ may not be regarded as fairly applying to systems of no degrees of freedom. We may therefore prefer to regard these equations as defining $\psi$ in this case.
2. See Chapter IV, page 35.
3. If the external coördinates determine a certain volume within which the system is confined, the contrary of (507) would imply that we could obtain an infinite amount of work by crowding an infinite quantity of matter into a finite volume.
4. It will be observed that, so far as the distribution is concerned, very large and infinite values of $\epsilon$ (for certain phases) amount to nearly the same thing,—one representing the total and the other the nearly total exclusion of the phases in question. An infinite change, therefore, in the value of $\epsilon$ (for certain phases) may represent a vanishing change in the distribution.
5. Compare Transactions Connecticut Academy, Vol. III, pages 116 ff.
6. See formula (517).
7. Strictly speaking, $\psi _{\rm {gen}}$ is not determined as function of $\nu _{1},\ldots \nu _{h}$, except for integral values of these variables. Yet we may suppose it to be determined as a continuous function by any suitable process of interpolation.
8. See page 184-186.
9. In this paragraph, for greater distinctness, ${\overline {\mathrm {H} _{\rm {gen}}}}|_{\rm {grand}}$ and ${\overline {\eta _{\rm {spec}}}}|_{\rm {petit}}$ have been written for the quantities which elsewhere are denoted by ${\overline {\mathrm {H} }}$ and ${\overline {\eta }}$.