106 Proceedings of the Royal Society of Edinburgh. {Sess. 
this, or a multiple of it, is readily supplied by the blows of other molecules. 
Hence the sum of E' and E" should remain equal to fRT, at least up to 
temperatures as high as are reached in experiments on the specific heats 
of gases. But the same result, as we have seen, follows from more 
ordinary dynamics. 
An apparently stronger case for the application of the Quantum Theory 
to the rotation of molecules in a gas is presented in the remarkable fact, 
discovered by Eucken* that when hydrogen is exposed to extreme cold 
(but is still a gas) its specific heat falls to a value which indicates that its 
molecules have ceased to take up energy of rotation. Eucken found that 
in hydrogen fell from 4*94 per gramme-molecule at 0° C. (approximately 
fR) to about 3T when the temperature was reduced below — 200° C. This 
has been confirmed by independent measurements of C P f and of y J. As 
the temperature falls, C v approaches a limit of f R, and y approaches a limit 
of If, showing that at a temperature of 30° or 40° absolute the molecule of 
hydrogen takes up energy of translation only, behaving as if it were a 
single-atom molecule of double mass. On the Quantum Theory this is 
accounted for by saying that though the molecule retains its “ dumb-bell ” 
form its energy-quantum with respect to rotation about a transverse axis 
is sufficiently large to exceed the energy of the blows which it receives at 
very low temperatures. Thus if we conceive the hydrogen molecule to be 
made up of a pair of masses, each equal to 1*62 x 10 -24 gr., at points 10~ 8 cm. 
apart, the moment of inertia I about a transverse axis is 8T x 10 -41 , and the 
energy-quantum A 2 /27r 2 I is 2'6 x 10 -14 ergs, when h is taken as 6’5 x 10 -27 
erg-seconds. To compare this with the energy that may be delivered by a 
blow, we have R for any gas equal to 1*35 x 10~ 16 per molecule, the gas-constant 
being T985 in thermal units per gramme-molecule, and Avogadro’s constant 
being 6*16 x 10 ~ 23 . Hence the average energy of translation per molecule, 
fRT, is 2'02 x 10~ 16 T, which becomes 5*5 x 10~ 14 ergs at 0° C. and 0*6 x 10 -14 
ergs at 30° absolute. The latter figure is well below the estimated value 
of the energy-quantum. So far as it goes, therefore, the general result of 
this calculation is in accord with the Quantum Theory. 
It is extremely difficult, however, to believe that a body of the assumed 
form, free to move and turn in space without constraint, and exposed to 
entirely casual blows from other like bodies — which may hit it anywhere 
and at any angle — can refuse to accept angular momentum from such blows 
* A. Eucken, Sitzungsb. d. k. preuss. Akad., Feb. 1912. 
t Scheel and Heuse, Sitzungsb. d. k. preuss. Akad., Jan. 1915 ; Ann. d. Physik, 1913, 
vol. xl, p. 473. 
f M. C. Shields, Phys. Rev., Nov. 1917. 
