219 
1921-22.] The Magnetic Character of the Quantum. 
classical theory and postulating a vibrator which does not lead to the law 
of equipartition of energy. 
Jeans, in his Report on Radiation and the Quantum Theory, says 
(p. 84) 
“ The new mechanics must differ from the old even as regards the motion 
of free electrons. For this reason it seems useless to attempt to explain 
away the conflict between the radiation laws and the classical mechanics 
by ingeniously devised special models of atoms, or special detailed mechan- 
isms of emission of radiation, which might seem, while obeying the classical 
laws, to give something approximating to Planck’s law.” “Any such 
attempt would first have to surmount the difficulty that any system what- 
ever, if it obeys the classical laws, must also in its state of thermodynamical 
equilibrium obey the law of equipartition of energy, which is known in 
turn to lead to Rayleigh’s radiation formula. And if this difficulty could 
be turned, as, for instance, by postulating a final state which was not one 
of thermodynamical equilibrium, the question of why all possible mechan- 
isms of radiation lead to Planck’s law, as they certainly appear to do, would 
remain untouched. And finally, if single electrons do not obey the classical 
laws, there would seem to be little gain in proving, if it could be proved, 
that complicated structures might possibly obey them.” 
Very similar was the position taken up by Poincare, who put forward a 
mathematical argument to show that not merely does Planck’s radiation 
law involve the hypothesis of discontinuities, but that the existence of 
quanta is necessitated by any law to which that of Planck might be re- 
garded as a first approximation. Poincare’s proof of the necessity of 
Planck’s hypothesis of quanta depends on the use of Fourier’s integral 
theorem to invert a particular infinite integral. It has been pointed out by 
Planck,* and also by Fowler,]- that there is a gap in Poincare’s argument 
due to the fact that the functions actually involved are such that Fourier’s 
integral theorem does not apply at all. On this ground Planck criticises 
Poincarb’s conclusion that the second form of the Quantum Theory — 
discontinuous emission and continuous absorption — is inadmissible, and 
suggests that it may be due to this gap in the mathematics that Poincare’s 
argument distinguishes between the first and second hypotheses, allowing 
the first to be necessary and the second impossible. Fowler, however, has 
shown that it is possible to make a simple extension of Fourier’s integral 
theorem so that the function required is properly catered for, and concludes 
that, if the groundwork of Poincare’s argument is admitted to be correct, 
* Planck, Acta Mathematica, vol. xxxviii, p. 38V, 1921. 
f R. H. Fowler, Proc. Roy. Soc., vol. xcix, p. 462, 1921. 
