bo 
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TRANSACTIONS OF SECTION A. 7 
7. On some Difficulties connected with the Kinetic Theory of Gases. 
By G. H. Bryan, Se.D. 
The recent attacks of M. Bertrand on Maxwell’s investigations emphasise the 
view that all proofs of the Boltzmann-Maxwell distribution involve some as- 
sumption or other, and that such assumptions are only justifiable in attenuated 
assemblages of molecules such as constitute an ideal gas. But if the thermal 
properties of gases are really due to molecular motions, as the kinetic theory sup- 
poses, the same must be true of the corresponding properties of matter in its other 
states; so that a kinetic theory of solids and liquids also must exist even though the 
complete investigation of that theory may present insuperable difficulties to the 
mathematician. Now, the most important physical property for which the kinetic 
theory has to account is that of temperature, and the existence of such a quantity 
depends on the fact that if a body A be in thermal equilibrium with B, and also 
with ©, then B will be in thermal equilibrium with C; in other words, the condi- 
tion of thermal equilibrium between A and B must be expressible in the form 
F(A) =f,(B) - - : ‘ . : (1) 
where the left-hand side involves no variables depending on the state of B, and 
the right-hand side involves no variables depending on the state of A. 
On the assumption that the temperature of a body is proportional to the mean 
kinetic energy of translation of its molecules, the condition of equal temperature 
requires that if the mean translational energies of two sets of molecules A and 
B are equal, no energy will be transferred from A to B. Now if we take 
only two molecules M and m, moving in the same straight line, the con- 
dition for no transference of energy between them is ot that their kinetic 
energies shall be equal. Indeed, Prof. Tait has shown that this condition holds 
good if the molecules of A and B are distributed according to the Boltzmann- 
Maxwell distribution, but not in general. 
The author is at present investigating what restrictions are imposed on the law 
of distribution of molecular velocity in order that the condition of thermal equi- 
librium may be expressible in the form (1), in other words, in order that tempera- 
ture may exist. The analysis is somewhat complicated, but it may be safely 
concluded, even at the present stage, that the existence of temperature cannot be 
inferred from dynamical considerations alone, independently of the law of dis- 
tribution. It will be necessary for us to regard the laws of thermodynamics as 
the fundamental assumptions of a general kinetic theory of matter rather than as 
the results to be proved, and we must therefore deduce from those laws the nature 
of the molecular motion which we call heat. 
MONDAY, SEPTEMBER 21. 
The following Papers and Reports were read :— 
1. On the Communication of Electricity from Electrified Steam to Air. By 
Lord Ketvin, /.2.S., Dr. Magnus Maciean, and ALEXANDER GALT. 
2. On the Molecular Dynamics of Hydrogen Gas, Oxygen Gas, Ozone, 
Peroxide of Hydrogen, Vapour of Water, Liquid Water, Ice, and 
Quartz Crystal. By the Right Hon. Lord Ketviy, G.C.V.0., P.B.S. 
In a communication, ‘ On the Different Crystalline Configurations possible with 
the same Law of Force according to Boscovich, to the last meeting (July 20) of 
the Royal Society of Edinburgh ‘a purely mathematical problem of fundamental 
importance for the physical theory of crystals—the equilibrium of any number of 
ints acting on one another with forces in the lines joining them—was considered 
in the simplest case of Boscovichian statics; that in which the mutual force 
between every pair of atoms is the same for the same distance between any two 
