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molecule of every one of the ordinary gases must have a vastly greater number of atoms in its constitution than those hitherto reckoned in regular chemical doctrine. Suppose, for example, there are forty-one atoms in the molecule of any particular gas; if the doctrine were true, we should have y=S9. Hence there are 117 vibrational freedoms, so that there might be 117 visible lines in the spectrum of the gas;
and we have k — 1 = “ ’0083. There is, in fact, no xJi\)
possibility of reconciling the Boltzmann-Maxwell doctrine with the truth regarding the specific heats of gases.
§ 27. It is, however, not quite possible to rest contented with the mathematical verdict not proven, and the experimental verdict not true, in respect to the Boltzmann-Maxwell doctrine. I have always felt that it should be mathematically tested by the consideration of some particular case. Even if the theorem were true, stated as it was somewhat vaguely, and in such general terms that great difficulty has been felt as to what it is really meant to express, it would be very desirable to see even one other simple caseT besides that original one of Waterston’s, clearly stated and tested by pure mathematics. Ten years ago*, I suggested a number .of test-cases, some of which have been courteously considered by Boltzmann ; but no demonstration either of the truth or untruth of the doctrine as applied to any one of them has hitherto been given- A year later, I suggested what seemed to me a decisive test-case disproving the doctrine; but my statement was quickly and justly criticised b}r Boltzmann and Poincare; and more recently Lord Rayleigh f has shown very clearly that my simple test-case was quite indecisive. This last article of Rayleigh’s has led me to resume the consideration of several classes of dynamical problems, which had occupied me more or less at various times during the last twenty years, each presenting exceedingly interesting features in connection with the double question: Is this a case which admits of the application of the Boltzmann-Maxwell doctrine; and if so, is the doctrine true for it?
§ 28. Premising that the mean kinetic energies with which the Boltzmann-Maxwell doctrine is concerned are time-integrals of energies divided by totals of the times, we may conveniently divide the whole cla^s of problems, with
* ■' On some Test Cases for the Maxwell-Boltzmann Doctrine regarding' Distribution of Energy/ Proc. Roy. Soc., June 11, 1891.
t Phil. Mag., vol. xxxiii. 1892, p. 350. “Remarks on Maxwell’s Investigation respecting Boltzmann’s Theorem.”
reference to which the doctrine comes into question, into two classes.
Class L Those in which the velocities considered are either constant or only vary suddenly—-that is to say, in infinitely small times—or in times so short that they may be omitted from the time-integration. To this class belong:
(a) The original Waterston-Maxwell case and the collisions of ideal rigid bodies of any shape, according to the assumed law that the translatory and rotatory motions lose no energy in the collisions.
(b) The frictionless motion of one or more particles constrained to remain on a surface of any shape, this surface being either closed (commonly called finite though really endless), or being a finite area of plane or curved surface, bounded like a billiard-table, by a wall or walls, from which impinging particles are reflected at angles equal to the angles of incidence.
(c) A closed surface, with non-vibratory particles moving within it freely except during impacts of particles against one another or against the bounding surface.
(d) Cases such as (a), (b), or (<?), with impacts against boundaries and mutual impacts between particles, softened by the supposition of finite forces during the impacts, with only the condition that the durations of the impacts are so short as to be practically negligible in comparison with the durations of free paths.
Class II. Cases in which the velocities of some of the particles concerned sometimes vary gradually ; so gradually that the times during which they vary must be included in the time-integration. To this class belong examples such as (d) of Class I. with durations of impacts not negligible in the time-integration.
§ 29. Consider first Class I. (b) with a finite closed surface as the field of motion and a single particle moving on it. If a particle is given, moving in any direction through any point I of the field, it will go on for ever along one determinate geodetic line. The question that first occurs is, Does the motion fulfil Maxwell’s condition (see § 18 above) ? that is to say, for this case, If we go along the geodetic line long enough, shall we pass infinitely nearly to any point Q whatever, including I, of the surface an infinitely great number of times in all directions? This question cannot be answered in the affirmative without reservation. For example, if the surface be exactly an ellipsoid it must be answered in the negative, as is proved in the following §§ 30, 31, 32.