384 TRANSACTIONS OF SECTION A. 
necessary that all the terms which are far advanced in the order or the series 
should be very small compared with some of those which precede them. It is 
therefore impossible for all the terms to be equal as required by the principle of 
equipartition. If, however, the principle of equipartition is limited by the 
principle of convergence, it yields some information as to the distribution of 
energy in the spectrum of a black body. In so far as it is legitimate to regard 
the infinite series, by way of approximation, as the sum of a large, but finite, 
number of terms, the principle of equipartition should be applicable, also as an 
approximation, and it yields Lord Rayleigh’s experimentally verified formula 
for the emissivity answering to long waves. ‘The principle of convergence 
shows, on the other hand, that for short waves the curve obtained by plotting 
emissivity against wave-length should fall towards the origin, as it is known to 
do. But this principle yields no information as to the position in the spectrum 
of the longest waves for which Lord Rayleigh’s formula fails to give a valid 
approximation. 
The arguments on which the theory of quanta was founded cannot be regarded 
as satisfactory. Indeed, the most convincing evidence in favour of the theory 
would seem to be the agreement with experiment of M. Planck’s formula, accord- 
ing to which the emissivity of a black body is given as a function of the wave- 
length A and the absolute temperature T, by an expression of the form 
Aa~5(eBAT—1)7', 
where A and B are properly determined constants. It may, therefore, be per- 
tinent to remark that from a mathematical point of view there must be infinitely 
many formule which would agree equally well with the experiments. In illus- 
tration of this statement, it may be mentioned that a formula proposed recently 
by A. Korn, according to which the emissivity of a black body would be given 
by an expression of the form 
Car energy 
where C and D are properly determined constants, when tested arithmetically 
over a wide range, yields results showing just about as good an agreement with 
the facts as Planck’s. It may be, however, that there is no simple formula, like 
those of Planck and Korn, which is applicable to all wave-lengths. However 
this may be, there seems to be no sufficient reason for regarding Planck’s 
formula as expressing a law of Nature. 
In further illustration of the contention that the resources of the ordinary 
theories are not exhausted, it may be pointed out that it is possible to extend to 
some additional cases the calculation, first carried out by H. A. Lorentz in the 
case of long waves, of the emissivity of a thin metal plate. He supposed the 
radiation to be generated in collisions between free electrons and atoms, and 
calculated the emissivity for waves of periods long compared with the times 
occupied in describing free paths. For this calculation he required to evaluate 
approximately a certain integral. Such an evaluation can be effected also in the 
case of waves which have their periods comparable with the times occupied in 
describing free paths, and, for a number of laws of variation of the acceleration 
during a collision, in the case of waves which have their periods comparable with 
the times occupied by collisions. As the wave-length diminishes the emissivity 
of the thin plate at first increases, then reaches a maximum, and finally diminishes 
to zero; as the temperature rises the wave-length answering to the maximum 
emissivity diminishes, as would be expected. But there is no simple analytical 
formula which represents the emissivity of the thin plate over the whole range 
of wave-lengths. 
Mr, Jeans: I do not think the mathematical question of the convergence of 
Professor Love’s infinite series need be considered at all. If the series is 
arranged in order of descending wave-lengths we may say roughly that the 
equations of the classical dynamics demand that the energy should be transferred 
along the terms of the series from left to right, and no steady state can be 
attained until the terms are all equal. The time taken for the energy to reach 
the more distant terms can be approximately calculated: before we have got 
very far along the series it becomes a matter of millions of years. Let us then 
