534 
NATURE 
4 
[Oct. 1, 1885 
of average free paths, so that it is difficult to imagine how 
any of the molecules in the more compressed gas can be said to 
be in the state, as to pressure, of the average molecules in the 
less dense gas. The free path of a molecule of the denser gas 
may at any instant be the same as the average free path of the 
molecules of the less dense gas; but its average free path will 
not be the same as theirs, and it is this that determines the 
pressure. In a system consisting of phosphorus and oxygen 
the possibility of chemical combination implies the possibility of 
an atom of phosphorus acquiring the same motion of translation, 
both as to speed and direction, as several atoms of oxygen, and 
of their jointly taking up the vibrational motions proper to an 
oxide of phosphorus at the temperature of the system, and that 
the transformations of energy involved in all this should be 
attended on the whole with a degradation. Since a diminution 
of the pressure of a gas means a degradation of its energy, this 
may facilitate combination when the mere fact of the molecules 
having instantaneous free paths of greater or less length would 
not suffice to produce such a result. 
Sir W. Thomson remarked that Boltzmann’s theorem was 
trae in one particular case, but a proof of this case could be 
arrived at without the aid of the theorem, so that this does not 
prove the truth of the theorem. On the other hand, he had 
never seen any reason for believing in it at all. If we take an 
absolutely elastic globe and cause it to rebound between two 
parallel absolutely smooth and hard planes in a region where 
gravity does not act, it will go on moving between the two. 
But he does not believe that this will continue for ever. The 
translational energy of the ball will get transformed into energy 
of higher and higher modes of vibration, so that at last the ball 
will come to rest, as it will be impossible for this energy to be 
retransformed into translational energy. 
Prof. J. J. Thomson said that he thought the reason that the 
ratio of the specific heats of a gas, as found by experiment, did 
not agree with the value given by Toltzmann’s theorem, was | 
because Boltzmann’s theorem was not true. 
Boltzmann, in his theorem about the distribution of energy in 
agas the molecules of which consisted of dynamical systems 
with 7 degrees of freedom, assumed that there were no limits 
to the velocity which any co-ordinates could have, and therefore 
that the limiting velocity which any co-ordinate could have was 
independent of the velocity of any of the others. Now it was 
easy to see that in some cases there must be limits to the 
velocities, for, take the case of a molecule consisting of two atoms 
attracting each other with a force varying inversely as the square 
of the distance between them, then, if the relative velocity ex- 
ceeded a certain value, the atoms would describe hyperbolas about 
their common centre of gravity, and the distance between them 
would increase indefinitely—in other words, the molecule would 
breakup. Again, if we considered the case of a series of balls con- 
nected together by springs and fastened to a system which vibrated 
much more quickly than the natural period of vibration of | 
the balls, then, if all the impacts fell on this systen, the 
dynamics of the case, as investigated by Stokes and Sir William 
Thomson, showed that any disturbance would not be equally dis- 
tributed among the balls, but that the energy in the balls would 
diminish in geometrical progression as we went away from the 
system at the end. It seemed, to say the least, rash in a case of 
this kind to assume that the velocity of any of the balls far 
away from the system was independent of those preceding it. 
He had devised a molecule which it was easy to see would 
not obey Boltzmann’s theorem. A was an envelope to the 
bottom of which a feeble spring was fixed, the other end of 
which was attached to a heavy weight, B. To this weight a 
strong spring was attached, to the other end of which a light 
weight, C, was fixed. 
such a length that it cnly extended beyond the envelope when 
the \prings were stretched. This system would have two 
\ 
\ 
periods of vibration—a quick one corresponding to the upper 
sphere, and a slow one corresponding to the lower one. Then 
if allthe molecules were stated, so that the amplitude of the quick 
vibration of C was much greater than the slow one, it was easy 
to see that the mean energy of the upper sphere would be 
greater than the mean energy of the lower ones, while, according 
to Boltzmann’s theorem, these two quantities ought to be the 
same. 
It might be mentioned that any co-ordinate which only entered 
the expression for the energy through its differential coefficient 
could be eliminated from the expression occurring in Boltz- 
mann’s theorem and the method applied to the remaining co- 
ordinates, so that, even if Boltzmann’s method was unobjection- 
able the result need not apply to co-ordinates of this kind. 
With regard to the second of the difficulties mentioned by 
Prof. Crum Brown, he thought that the point raised presented 
no difficulty if we took Williamson and Clausius’s view of 
chemical combination. According to this view it was necessary 
to consider the number of molecules dissociated as well as the 
condition of the molecules ; and though, if we took two gases at 
any temperature, it was true that there were a finite number of 
their molecules whose energy did not differ much from the 
mean energy of the molecules at the temperature at which these 
combined, yet it did not follow that a finite proportion of these 
were dissociated, and if there were not we could not expect 
them to combine. If the collision between two molecules in 
nearly the same condition was more efficacious in splitting up 
the molecules into atoms than a collision between molecules in 
widely different conditions, then we should not expect a finite 
proportion of the molecules in any state widely different from 
the mean to be dissociated. 
Prof. W. M. Hicks said that one of the greatest objections to 
Boltzmann’s theorem appeared to him tobe the difficulty in believ- 
ing that the mean energy ofany vibration whatever of an atom was 
susceptible of unlimited increase, and referred to the case of a 
vortex ring inside a rigid spherical shell, where such ener, 
could not be made to exceed a particular limit. Asa matter of 
fact it was not proyed that Boltzmann’s theorem must correspond 
to the actual state, but only that an arrangement given by his 
theorem, if a possible one, was a permanent one. He stated 
that if the momenta could not exceed definite limits, Watson’s 
proof could easily be modified to show that the energy was not 
distributed equally amongst the degrees of freedom. On the 
other hand, it was not permissible to assume all momenta con- 
sistent with the equation of energy as existent. As an example, 
the case of a system of mutually attracting spheres might be 
taken. Here the equation would admit of the infinite velocities 
due to infinitely near approach of the centres, which would in 
the actual case be prevented by the finite size of the spheres. 
Further, any particular system might possess other integrals of 
the equations of motion, which would introduce further limita- 
tions. 
Prof. Osborne Reynolds remarked that the kinetic theory is 
only supposed to be true in as far as the assumptions on which it 
was based represented the actual circumstances. In these 
assumptions no account whatever was taken of any resistance to 
which the molecules in their motions might be subjected, other 
than that which arose from the mutual encounters. Whereas it 
was perfectly well known and certain that there must be such 
resistances connected with the radiation of heat—these resist- 
ances, applying only to motions of certain character, z.e. to the 
vibratory motions, whatever these may be. Neglecting these 
resistances, the kinetic theory points to the conclusion that the 
| mean energy in each one of these vibratory motions would be 
the same as in each one of the translatory motions. In the same 
way, neglecting resistance, a pendulum continuously struck at 
varying intervals with a hammer of a given weight and moving 
| ata given speed would possess the same mean energy whether 
the intervals were to be measured by years or seconds. But 
| experience at once showed that with friction, the shorter the 
| interval between the blows and the smaller the friction, the 
greater would be the mean energy of the pendulum. So, 
taking resistance into account, it would follow from the kinetic 
theory that the mean energy in the so-called degrees of freedom 
| would be greatest in those in which the diffusion of energy was 
A rod of small mass was fastened to C, of | 
greatest and the resistance least, while it would be least in those 
in which the rate of communication was least and the resistance 
greatest. Hence, in any gas, the mean energies of translation, 
in which there is most rapid communication and no appreciable 
resistance, will be much greater than the mean energies of 
—_” 
