It appears that the metal by its own attraction for the charge on the ion tends to neutralize the effect of the velocity of impact; so that, with the current in the steady state, the metal exerting the greater attraction for the charge it receives possesses the smaller drop of potential. The series of metals arranged according to the magnitude of the cathode drop would be thus the reverse of that for the anode drop.

We should expect the relatively large variation in both cathode and anode drops when the electrodes become tarnished to be due to the same influences; but here the results are exactly the reverse of what we should predict from the contact-potential series. According to Lord Kelvin "dry oxide of copper is resinous to copper in contact with it, and dry oxide of zinc is resinous to zinc in contact with it, just as copper is resinous to zinc in contact with it." While we would predict then, that, by tarnishing, the drop at the anode should decrease, I have found by experiment (1. c.) that there is invariably an increase in the drop at the anode, Warburg (l. c.) having previously found that the drop at the cathode decreases.

Physical Laboratory,

University of Nebraska, Lincoln.

LX. The Theoretical Evaluation of the Ratio of the Specific Heats of a Gas. By J. H. JEANS, B.A., Fellow of Trinity College, and Isaac Newton Student in the University of Cambridget.

Introduction.

THE Maxwell-Boltzmann theorem on the Partition of Energy leads, as is well known, to a value of y which is not in accordance with observed values. In a recently published paper ‡ I have suggested that an escape from the conclusions of this theorem might be found by taking account of the interaction between matter and æther; and it was shown that even a small interaction might entirely modify the conclusions of the theorem. In the present paper an attempt is made to work out in greater detail the nature of these modifications, and, in particular, to examine whether it is possible, in this way, to obtain a theoretical evaluation of y which shall be in agreement with observed values.

γ

• Lord Kelvin, Phil. Mag. July 1898, p. 82.

+ Communicated by Prof. J. J. Thomson, F.R.S.

64

+ The Distribution of Molecular Energy," Phil. Trans. cxcvi.

p. 397.

Experimentally the value of y is best found indirectly by observation of the velocity of sound; and this introduces a further complication into the question. This complication arises as follows.

In arriving at Laplace's value for the velocity of sound in a gas, it is assumed that the ratio of the two specific heats of a gas has a definite value, y, which depends solely upon the nature of the gas. This, as can easily be verified, amounts to assuming that the mean internal energy of a molecule bears, at every instant, a constant ratio -1 to the mean energy of translation.

2
3(7-1)

Now, after the lapse of infinite time, a gas tends to assume a steady state in which the two mean energies may, perhaps, be legitimately supposed to bear to one another this constant ratio; but the case is different with a gas in which the value of one of the quantities in question is continually caused to vary owing to the passage of a wave of sound. For the mechanism by which the balance of energy is adjusted cannot be supposed to work with infinite rapidity, so that the ratio in question will never have the actual value which must be assigned to it in order to arrive at the Laplacean velocity of sound. The question as to whether or not this want of steadiness in the ratio of the two energies is sufficient to influence appreciably the transmission of sound, is therefore seen to be one requiring investigation.

General Theory.

§ 2. Let us, for the sake of simplicity, consider a gas of which the molecules possess only one kind of freedom in addition to the freedom to move in space. Let the mean

energy of translation in space be denoted by E, and the mean total energy by E+F, the energy F arising from the remaining kind of freedom, which may be either a rotation or an internal vibration.

The quantities E and F will be capable of variation owing to the transfer of energy which is effected by collisions between the molecules of the gas. In order to obtain sufficiently general results we find it necessary to imagine a second path for the transfer of energy, namely æther-vibrations. Regarding the molecule as an electromagnetic system, the vibrations or rotation of the system will send out electromagnetic waves into the æther, and collision of the two molecules will also send out a system of irregular electromagnetic waves or

* Lord Rayleigh, 'Theory of Sound,' § 246.

"pulses." The energy of these waves will be either partially or entirely absorbed by the other molecules of the gas, either by forcing vibrations in them, or by affecting their velocities of translation or rotation; and in this way we have a second path for the transfer of energy. If this absorption is complete, the total mean energy E+F will remain constant throughout, and a steady state will be possible. If the absorption is only partial, a steady state will only be possible if we suppose energy introduced into the gas by some external agency, of amount sufficient to compensate that which is lost by the radiation of waves.

If the law of distribution of E and F among the molecules of the gas is known, it will be possible to calculate the two transfers caused respectively by collisions and electromagnetic waves. The rate at which F increases owing to collisions will be some function of E and F, which will for the present be denoted by (E, F); the rate at which F increases owing to radiation and absorption may similarly be taken to be (E, F). We therefore have, for the total change in F,

There must be at least one steady state; let the corresponding root of

be

$(E, F) +↓(E, F)=0.

F=ƒ(E).

(2)

(3)

When the reaction with the æther is of zero amount, equation (3) will reduce to the expression of the MaxwellBoltzmann law; but a very small reaction may, as was pointed out in the paper previously referred to, suffice to alter entirely the form of this equation. The value of y is now given by

where f'(E) stands for df(E)/dE.

This value of y is not restricted to being one of the series given by the formula 1+ 2/n, and is, moreover, capable of variation with the temperature.

§ 3. The right-hand member of equation (1) contains as a

factor either F-f(E), or else some power of this factor, or else some other function of this factor which vanishes when

the factor is put equal to zero. In order that a simply periodic variation in F and E may be possible (i. e. in order that sound may be propagated in the gas) the first of these alternatives must be the true one. In this case, assuming the values of E and F to vary only slightly from their equilibrium values Eo, Fo, equation (1) may be written in the form

where x(E) is some function of E, which is positive if the gas tends to return to the steady state in which EE, F=F. Assuming as a general solution

E=E+ Eeipt,

F=Fo+F1eipt,

we find from (5) the equation

ipF1=-x(E6) [F1 −ƒ' (E) E1];

§ 4. Let, p be the pressure and density of the element gas under consideration, and let us assume as a solution the values

of

for

The quantities, p, and E are connected by the relation

where is a constant. Substituting the assumed values of w, p, and E, we find at once the relation

We further suppose that no heat enters or leaves the element in question, so that the total energy of the element

remains constant. This gives the equation *

3 (8E+SF)+pEd (2) = 0,

in which & denotes excess above the mean value. Substituting for E, F, and P, this gives

Combining this with equation (11), we find

If we had supposed the gas to obey the ordinarily-assumed equation for adiabatic motion, viz.

§ 5. Comparing this with equation (12), we see that the quantity I given by equation (13) is a generalized form of the usual quantity 7. When the variation in E is infinitely slow (i. e. when p=0), the ratio between F and E is always the equilibrium ratio, and I becomes identical with Y. Šo that we find, in this case, that equations (14) and (5) become identical. When, on the other hand, the variation in E is of infinite rapidity, the value of F is unaffected by variations in E, and maintains a constant value corresponding to the mean value of E. For this reason, the gas will behave like a gas of which the molecules possess only the three translational degrees of freedom, i. e. like a gas for which y=13.

Boltzmann, Vorlesungen über Gastheorie, i. p. 56.