378 TRANSACTIONS OF SECTION A. 
We accordingly see that the average energy of a vibration in a gas is RT, 
HH 
while that in ether or a solid is RT x — It will, of course, be remarked 
er 
that the latter formula reduces to the former on putting «—0. It may further 
be noticed that the values of » involved in the gas problem are very small, 
compared with those involved in the other problems, so small, in fact, that x 
is negligible, and the formula (3) becomes indistinguishable from RT. 
Knowing the average energy of each vibration in a medium, it may or may 
not be possible to deduce some information as to the mechanism by which this 
partition of energy is produced. In the special case in which the partition of 
energy is such that each vibration has energy RT x sy the problem has been 
solved with great completeness by Poincaré. The quite definite result is obtained 
that, to arrive at this particular law of partition of energy, the exchange of 
energy between matter and ether must take place by finite jumps of amount 
e, given by e=hv, This is, of course, the hypothesis, spoken of briefly as the 
quantum-hypothesis, which was first suggested by Planck, and from which his 
radiation formula (1) was first deduced. It now appears from the work of 
Poincaré that no other hypothesis could have led to Planck’s formula; thus the 
x 
result of experiment leads inevitably to the law RT xX 
ee 
i? and this in turn 
leads inevitably to the quantum-hypothesis. 
This result is so complete, so definite, and above all so revolutionary, that it 
will naturally be most closely scrutinised. We shall probably feel inclined to 
trust to the accuracy of Poincaré’s mathematics, and examine the physical 
assumptions on which the result is based. They are two in number. First is 
the assumption that the average energy of a vibration in matter is the same as 
that in ether—in other words, the assumption that formule (2) and (3) are 
identical. Poincaré based this assumption on a theoretical proof by Planck, but 
since he wrote the more direct evidence of Debye’s work has appeared, and may 
be considered to have put the matter beyond doubt. Next comes the assump- 
tion that formule (2) and (3) represent a real final state of thermodynamical 
equilibrium, and this assumption is perhaps slightly more open to question. 
Speaking for myself, I may perhaps be allowed to say that I have devoted 
several years of work to an attempt, quite unfruitful as it turned out, to recon- 
cile the laws of radiation with the classical mechanics by assuming that formule 
(1), (2), and (3) do no¢ represent a real final state. The more one works on this 
assumption the more one is forced to realise that all the facts are against it; the 
classical mechanics, coupled as they must be with this assumption, really show 
no power of explaining the facts of radiation; the new mechanics, based on the 
quantum hypothesis, show just that power of explaining and predicting facts 
which is to be expected of a new truth in its infancy. 
Some of the more conservative of us may feel tempted to challenge the 
accuracy of the law (2, 3) on which Poincaré’s result is based, but this possibility 
has been foreseen by Poincaré. He proves, in his own words," 
‘L’hypothése des quanta est la seule qui conduise 4 la loi de Planck.’ 
‘Quelle que soit la loi du rayonnement, si l’on suppose que le rayonnement 
total est fini, on sera conduit 4 une fonction présentant des discontinuités 
analogues 4 celles que donne ’hypothése des quanta.’ 
There is no escape, then, by appealing to errors of observation; a discon- 
tinuous system of mechanics is in any case thrust upon us, and all that an appeal 
to errors of observation could possibly do would be to modify slightly the laws 
governing the system of discontinuities. 
One special importance of Poincaré’s result is that it divides all possible views 
and theories of radiation into two sharp classes. Every view or theory must 
logically involve either the belief that Poincaré is wrong, or the belief that he is 
right, together with all that this involves. His work has shown that there is no 
middle way. For myself I feel logically compelled to accept the quantum- 
hypothesis in its entirety. 
But logical necessity of this kind is made mentally more palatable if we can 
1 Journ. de Phys., January 1912, pp. 37, 39. 
