196 Proceedings of the Royal Society of Edinburgh. [Sess. 
can in fact be carried out in a somewhat modified manner. But this con- 
struction then forms no objection to Boltzmann's general line of thought. 
§ 1. Boltzmann has given, in §§ 74-86 of the second volume of his 
Gastheorie, on the basis of the H-theorem, the theory of the equilibrium of 
heat in multiply-atomic gases. We shall frequently refer in what follows 
to this presentation. Boltzmann makes the following remarks in preface 
to the deduction of the law of distribution (page 241): — “The complete 
development of the proof that the distribution, e -/l ‘ Energy , is the only 
possible one which can maintain itself in a stationary gas, whose molecules 
are multiply-atomic, has not hitherto been successful in complete generality. 
Yet the development of this proof is possible, in the simplest and practically 
most important cases, to the same extent as for mon-atomic gases.” 
The deduction of the distribution from the H-theorem involves two 
distinctly separate steps : — 
A. The deduction of the known equation 
(see §§ 74-82). 
B. The solution of this (functional) equation. Whilst this second step 
is very simple in the case of monatomic molecules, it offers special difficulty 
in the case of multiply-atomic molecules. (For the discussion of particular 
cases, see Boltzmann in §§ 83-86.) But, at the same time, this second step 
exhibits the possibility, e.g. through special assumptions regarding the 
nature of the collisions between the molecules, of finding cases of non- 
equipartition. 
§ 2. Each molecule is made up of atoms A, B, C . . . . With reference 
to the nature of the collisions between the molecules, Boltzmann follows 
out twa essentially different assumptions. 
Assumption I. — Every atom of every molecule can collide with every 
other (§§ 83-84). 
Assumption II. — In each molecule the atom A alone is susceptible of 
collision ; the remaining atoms are influenced only through the changes of 
motion of the atoms A. 
In the case of assumption I., the H-theorem furnishes a complete estima- 
tion of the speed distribution for atoms of all kinds. As well the atoms 
A, as B, as C, .... , obey Maxwell’s speed distribution, and certainly are 
such that the mean kinetic energy (time, and number, mean), for the atoms 
of all kinds, has the same value (p. 243). 
In the case of Assumption II., on the contrary , the H-theorem furnishes 
too few equations, apart from supplementary assumptions , for the deter - 
