374 Proceedings of the Royal Society of Edinburgh. [Sess. 
The following inequalities must hold good : — 
m n 2 
r pah 
> m J 
i — 
i 
1 
\dxj _ 
A 
_\dx 2 J 
and 
03q 
dx Y 
p, y 
< 
> 
d%V~ 
dx 2 / _j 
d&j 0<£ 2 
_dx 1 dx 2 _ 
ra n 
/0# ,\ 2 1 
r0<L 0<Ln 
_ 
+ m 2 
_dx 1 dx 2 _ 
> 0 , 
where it is supposed, for convenience, that m 1 is the greater of the 
two masses. 
Conclusions. 
(20) A purely dynamical problem has been considered, which may throw 
additional light on the theory of gas molecules. 
Equations have been obtained which should determine the law of partition 
of energy for a system composed of two attracting particles moving in any field 
of force, when certain hypotheses concerning the probable motion have been 
made ; these hypotheses require that the mean value of V 2 V should be 
positive, where V is the potential energy due to all the forces of the system ; 
so that the field has been assumed not to be a Newtonian field. There are 
three such equations, which I have called, for obvious reasons, the equations 
of translation, vibration, and rotation. 
From these equations I have, by equating to zero the three component 
mean energy accelerations, obtained three equations of energy equilibrium 
— in other words, equations which should determine the mean values of the 
squares of the velocities of the system corresponding to a stationary state ; 
but a term involving the mean value of the square of what I have called the 
rotational energy appears to be an unfortunate hindrance to the work in the 
general case. 
A condition has been obtained that Maxwell’s Law of Equipartition 
of energy should hold good, but there is apparently no reason why it should 
be satisfied. This condition is quite independent of the masses of the 
particles, and becomes a very simple one when the law of force between the 
particles is that of the inverse square of the distance. 
In none of the special cases considered is equipartition found to 
necessarily exist ; and in the particular case in which the attraction between 
the particles offers very great resistance to any variation of the distance 
between them the Law cannot possibly hold good. 
