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condition that if the kinetic energy be reduced to a sum of n squares, the mean values of these squares are equal. Hence the conclusion may be stated thus::

If the necessity for making Q/T a complete differential can be established as a substantive law by independent evidence, the investigation affords an independent proof of Maxwell's law of partition of kinetic energy between the momentoids for such a system.

Conclusion.

60. The conclusions arrived at in the present Report are to be regarded as superseding the statement of Clerk Maxwell's Theorem in §40 and the greater part of §§ 41, 42, 43 of Section III. of my first Report. The rest of that Report is not, so far as it goes, materially affected by any results now established, although several important questions connected with the Boltzmann-Maxwell Law have now received a definite answer.

The proof of the law and the assumptions involved in it are fairly satisfactory for gases whose molecules collide with each other to a certain extent at random, but in a medium in which the molecules never escape from each other's influence the subject still presents very great difficulties.

Even should it be shown that the law cannot be disproved for such a medium, there still remains the question as to whether the distribution is the unique one satisfying the conditions of permanence. The general question of uniqueness, even in some of the cases where the law admits of more or less satisfactory proof, still suggests some questions for investigation. Intimately connected with this is the difficult question of stability. For example, when a gas is condensed, its density at any point at first remains proportional to e" in accordance with the Boltzmann-Maxwell Law; but when a certain stage is reached, instability sets in, and part of the gas liquefies. If the Second Law be true, the new distribution satisfies Maxwell's law of partition of energy. Does it likewise satisfy the Boltzmann-Maxwell Law?

The connection with the Theory of Probability still suggests subjects for research. The relations of electrical and optical phenomena to the Kinetic Theory open up an almost unexplored field.

61. It only remains for me to thank all those who have assisted me in collecting materials for this Report. I am particularly indebted to Dr. Ludwig Boltzmann for his kindness in sending me copies of nearly all his writings, and for several valuable suggestions that have helped to clear up difficulties in the work. My thanks are also due to Mr. Burbury, Dr. Ladislaus Natanson, Professor Sydney Young, and others for similar help, which has very materially lightened the work of consulting and examining the large mass of existing literature relating to this most interesting branch of Mathematical Physics.

APPENDIX A.

The Possible Laws of Partition of Rotatory Energy in Non-colliding Rigid Bodies.

The motion of a rigid body about a fixed point or about its centre of mass under no forces affords one of the best test cases bearing on Maxwell's law of partition of kinetic energy.

The equations of motion referred to the principal axes are of the form

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1, 2, 3 be the initial angular velocities about the principal

Let axes, and let

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since each of the two determinants has two rows or columns equal. Therefore

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Since T constant for any body, it follows that if a very large number (N) of such rigid bodies have their angular velocities initially so distributed that the number with angular velocities between 1 and 1+d1, 2 and N2+dN2, N3 and 3+d, is

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the distribution of any subsequent time will be given by

Nƒ(T) dw¡dwdwz .

and therefore distribution will be permanent.

(69)

With this distribution it is easy to see that at any instant the average values of

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over the different bodies are equal to one another, so that Maxwell's law of partition of kinetic energy is satisfied.

But the equations of motion have a second integral expressing the constancy of resultant angular momentum, namely,

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will (in the absence of collisions between the bodies) be permanent.

And it can now be shown precisely in the same way as before that the mean values of

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are now inversely proportional to A, B, C, and Maxwell's law of partition of energy does not hold good.

It is only when the distribution of velocities or momenta is determined by a function f of the energy alone that we can assert that the mean values of the different squares forming the kinetic energy are equal.

APPENDIX B.

On the Law of Molecular Distribution in the Atmosphere of a

Rotating Planet.

Suppose a mass of gas-molecules to be situated in a field of force that is symmetrical about a fixed axis-for example, the field due to the attraction of a spheroidal planet.

Let this axis be chosen as the axis of z, and, in the first place, let the molecules be monatomic, so that they may be regarded as smooth homogeneous spheres or material points.

Let m, M be the masses of two molecules, Em, EM their total energies, P, P their angular momenta about the axis of z, so that

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Then, since the field is symmetrical, the angular momenta p, P are constant in the absence of collisions, and at a collision the total angular momentum is unaltered, so that

p+P=p'+P'

Also similar relations are satisfied by the energies Em, Ex

Therefore, if the distributions of co-ordinates and velocities be given by the expressions

fdx dy dz du dv dw and F dX dY dZ dỤ dV dW,

the conditions of permanence between collisions, and the functional equation for collisions

are satisfied by

f=n exp

ƒFƒ F

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where h, k are any constants whatever. Putting k=h, we have, if x be the potential energy of m,

f=n exp -h (T—Np+x)

an exp-hm (u2 + v2+w2)—mn (vx— uy)+x}

=nexp―h {} m [(u+Ny)3 + (v−Nx)2+w2]+x−}mN2 (x2+y2)} (73)

Take axes of E, n, rotating about the axis of ≈ with angular velocity and instantaneously coinciding with the axes of x, y, z.

Then the relative velocities of a molecule referred to these moving

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Hence the distribution may be written

n exp−h {}m (¿2+ñ2+¿2)+x−§mN2(¿2+n2)} . di dŋ dÿ dź dŋ dỷ. (74) Therefore the velocities relative to the moving axes follow the Boltzmann-Maxwell distribution, and in addition to this the molecules have a superposed motion of rigid-body-rotation with angular velocity . And the density at any point is the same as if the gas were acted on by 'centrifugal force' having a potential -2 (2+n2), and the reversed angular velocity were applied to every molecule.

-

Hence the Kinetic Theory may be applied to the atmospheres of planets by reducing the planets to rest and applying centrifugal force to the atmospheres in the usual way.

It is interesting to notice that the temperature 3/2h is the mean kinetic energy of the translational motion relative to the planet and not the total mean translational energy.

The results can evidently be generalised for the case when the molecules are rigid bodies of any kind. Let w,,, w, be the angular velocities of such a molecule about axes through its C.M. parallel to the axes of x, y, z, and let w1, w2, 3 be its angular velocities about its principal axes. By Appendix I., dw, dw, dwz is independent of the time, and evidently

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n exp —h (T—Op+x). dx dy dz du dv dw dw, dw, dw,

Now if A, B, C, D, E, F denote the moments and products of inertia, the kinetic energy of rotation of the molecule is

T={(A, B, C, — D, -E, -FX w., wy, w;)2

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T―Op+x=&m [(u+Ny)2 + (v−Nx)2+w2]—§mN2 (x2+y2)
+ 1⁄2 (A, B, C,—D, —E,-FX, w, w—N)2 — } CN3 +x

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f=n exp―h {}m (¿2 +ǹ2 +¿a3) + § (A, B, C, ‒ D, - E, -FX w2, w, w;)2

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where

T, is the kinetic energy of the relative motion n;
x is the potential energy due to the field;

V1 is the potential energy due to centrifugal force.

Hence, as before, the Boltzmann-Maxwell Law holds for the system obtained by applying the reversed angular velocity and the centrifugal force whose potential is - (+ y2) at every point of the gas.

It would not be difficult to extend the proof to the case of a rotating ellipsoidal planet with three unequal axes, where the field of force is not symmetrical about the axes of rotation, but the investigation would hardly be sufficiently interesting to be worth giving in detail. It will also be admitted, without difficulty, that similar conclusions must hold good when the planet and atmosphere besides rotating have a common motion of simple translation.

In a communication read at the Nottingham meeting of the Association I worked out certain results of applying the Boltzmann-Maxwell Law to the atmospheres of planets; but in these calculations no account was taken of axial rotation, as I did not at that time see how the effect of this rotation could be determined. The numerical results there obtained hold good, without modification, at points along the polar axes of the various bodies considered. The effects of centrifugal force on the distributions now furnish a promising subject for future investigation, about which I hope to say more shortly.

APPENDIX C.

On the Application of the Determinantal Relation to the Kinetic Theory of Polyatomic Gases. By Professor LUDWIG BOLTZMANN.

We shall consider a gas whose molecules are compound (or polyatomic), but are all similarly constituted. Let a, b, c, be the co-ordinates which determine the position and configuration of a molecule of such a gas; and let p, q, r,... be the corresponding momenta. Let us suppose that the time during which any one molecule acts upon or is acted upon by other molecules is short in comparison with the whole time of its motion. Let the gas be contained in a vessel of invariable form. After a certain time the state of the gas will become stationary, and the question is, what is then the probability that the co-ordinates and momenta of any one molecule lie between certain limits? To express the probability by means of a number let us suppose the stationary state to last for a long time, . Divide this time into n infinitely small parts, 9. We shall call The Moon's Atmosphere and the Kinetic Theory of Gases,' Nottingham Report,

682.

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