476 
PROFESSOR J. J. THOMSON ON SOME APPLICATIONS OF 
where (f> is given by the equation dcf) = dQ/T m . Making the first supposition, and 
remembering that the absolute temperature measures the mean energy due to the 
translatory motion of the molecules, we see that it is equivalent to supposing that the 
mean total kinetic energy of the molecules is a constant multiple of their mean trans¬ 
latory energy. This is the assumption which was originally made by Clausius, and 
we see that it must be made if we are to derive the Second Law of Thermodynamics 
from the principle of Least Action. 
Boltzmann, in his celebrated investigation'" of the distribution of energy among 
the molecules of a gas, each molecule of which possesses n degrees of freedom, arrives 
at a much more definite result. According to this investigation the mean kinetic 
energy corresponding to each degree of freedom is the same, so that the mean kinetic 
energy due to the translatory motion of the centres of gravity of the molecules is 
only 3 /n of the mean total kinetic energy of the molecules. The proof of this 
theorem given by the author seems to me to be open to grave objection, and the 
results to which it leads have certainly not been reconciled with the properties 
possessed by actual gases. According to this theorem, the result is the same, 
whatever be the constitution of the molecule, and whatever the forces exerted by one 
molecule on another when they come so close together as to be within the range of 
each other’s action. Boltzmann shows that, if the number of molecules which have 
the coordinates q } , q. 2 , . q n and the corresponding momenta p x , p 2 ,. p n 
between the limits q x , q x + Sgq, q 2 , q 2 -f Sq 2 , . . . q n , q n + Sq H , p 1} p x + 8 p lf . . . 
Pm Pn + Sp Hf is 
(y,-k(T + x ) dq^ dq 2 . . . dq n . dp x dp. 2 . . . dp n , 
where C and h are constants, and T and y the kinetic and potential energies of such 
a molecule; then the number of such molecules will remain constant, as in a given 
time as many molecules pass out of that state as enter it. Thus, if this distribution 
is ever established, it will be a steady distribution, i.e., the state of the gas will not 
change. To prove the theorem we have quoted above, Boltzmann integrates this 
expression, assuming that each velocity may have all values from plus infinity to 
minus infinity. It seems to me, however, that this assumption is not legitimate, 
and that before we can fix the limits of the velocity we must know the nature of 
the molecule and the forces between two molecules when they come within the sphere 
of each other’s action. We can easily imagine cases in which the assumption is not 
true. Take, for example, the case of two bodies describing orbits about their centre of 
gravity under each other’s attraction. If the relative velocity of the bodies exceeds a 
certain value, which depends upon the distance between them and the law of 
attraction, the two bodies will not remain together, but will separate until they are 
finally at an infinite distance apart. Thus, if the two bodies represent the atoms in a 
* Boltzmann, ‘ Sitzungsberichte der Kaiserlichen Akademie der Wisseuschaften,' vol. 63 (Abth. 2), 
Wien, 1871. See also Maxwell, 1 Transactions of tbe Cambridge Philosophical Society,’ vol. 12, 1879. 
