382 TRANSACTIONS OF SECTION A. 
that cannot be called very small. It may be added that, according to Maxwell’s 
equations of the electro-magnetic field, an initial disturbance of equilibrium must 
always be propagated over a continually increasing space. 
We might now suppose that the exchange of energy between a vibrator and 
the ether can only take place by finite jumps, no quantity less than a quantum 
being ever transferred to the medium or taken from it. Something may be 
said, however, in favour of the opposite hypothesis of a gradual action between 
the ether and the vibrator, governed by the ordinary laws of electro-magnetism. 
Indeed, it has been shown already, in Planck's first treatment of the subject, 
that by simply adhering to these laws, one is led to a relation between the 
energy of the vibrator and that of the black radiation, of whose validity we 
have no reason to doubt. 
The problem solved by Planck admits of a wide generalisation. Instead of 
his linear vibrator, we may conceive a body having any number of degrees of 
freedom and capable therefore of vibrating in many different fundamental 
modes. If such a body carries some distribution of electric charges and is 
surrounded by black radiation, it will be set in motion in all its modes, and the 
energy of each vibration is found to be 
Af 
Sr f(A), 
where \ is the wave-length and 
{{A) da 
the energy of the black radiation per unit volume, belonging to rays with wave- 
lengths betweenNandA + dA. But, by Planck’s equation, which we may use as 
an empirical formula representing the actual distribution of energy, 
{(A) =8"RTaA~4, —* 7 
where x has the same meaning as in Mr. Jeans’ formule, so that the energy of 
the body for each normal vibration becomes 
BUTE eee ty 
e—] 
This result shows the intimate connection between two of the expressions given 
by Mr. Jeans, and it justifies Debye’s method for calculating the specific 
heat of a solid body. The important point is that it can be obtained without 
the introduction of anything beyond the ordinary electro-magnetic equations. 
So far, the hypothesis of quanta is found to be unnecessary or even inad- 
missible, and perhaps the best course we can take will be to limit it to that part 
of the theory in which we consider the equilibrium between the vibrators and 
the material particles. It is possible that we shall be able after all to work 
out a satisfactory theory on the basis of certain assumptions involving a dis- 
continuous transfer of energy between these two parts of the system. Of 
course, when we try to do so, it may be very helpful to have before us some 
special model of the structure and properties of a vibrator, such as that which 
Sir J. J. Thomson brought forward yesterday. Some years ago, Dr. E. A. 
Haas, of Vienna, proposed another interesting model, which, however, does not 
illustrate an important feature that is explained by Sir J. J. Thomson, viz., the 
proportionality between the magnitude of a quantum and the frequency of 
vibration. 
Following Mr. Jeans, I must finally say some words on the equilibrium 
between the radiation and a system of free electrons. Starting from Drude’s 
theory of the metallic conduction of electricity, I once calculated both the 
absorption and the emission of rays by a thin sheet of metal. By combining the 
two results I obtained an expression agreeing with Lord Rayleigh’s radiation 
formula, which, as is well known, holds for low frequencies. My calculations 
had been expressly limited to these, and I therefore hoped for some time that 
it would be possible to deduce Planck’s equation by leaving aside the restric- 
tion, going more deeply into the details of the problem. At present we may be 
a 
