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LXIX. JVote on certain Dynamical Analogues of Temperature 
Equilibrium. By G. H. BfiYAN (Bangor)*. 
IN the Archives JVeerlandaises for 1900 (Livre jubilaire 
dedie a M. H. A. Lorenfz) I described under the title 
of " Energy Accelerations " a method o£ studying problems 
dealing with the partition of energy in systems of particles in 
which some kind of statistical equilibrium exists. This 
method consists essentially in writing down expressions for 
the second differential coefficients with respect to the time of 
the squares and products of velocities or momenta of the 
system considered. 
The method in question appears to have been previously 
employed by Mr. Burbury, who, however, did not employ the 
term " energy-accelerations " in connexion with the second 
differential coefficients in question. 
I was, and still am, of opinion that a further study of 
energy-accelerations must have an important bearing on all 
problems in the Kinetic Theory dealing with questions of 
temperature-equilibrium, and the fact that no attempt seems 
to have been made to follow T the problem up, may be taken as 
evidence of the large congestion of unsolved problems which 
presses heavily on the shoulders of the mathematician of 
to-day. The present note deals with two simple applications 
which happened to remain unnoticed when the paper was 
written. 
Consider in the first place an ideal medium formed of 
material particles uniformly distributed in space, both as 
regards position and as regards direction, and attracting or 
repelling one another according to any law of force as a 
function of the distance between them. 
If we confine our attention to one particular particle, the 
effect of the other particles will be to produce varying fields 
of force acting on the particle in question. Take now the 
equation of energy-accelerations which may be written in 
the more general form : — 
d 2 ri 2 . l/dV\ 2 <PV 2 d 2 V d 2 V d*V 
— i±mu ) = — - - - ) —u -~- u l -r-^r — uv-, — : nic - n — =- 
dt- K - m \ dx) dt dx dx 2 dx dy dx dz 
where ?/, v, ic stand for velocity components, V the potential 
of the field due to the other particles. 
The assumption that the field produced by the other 
particles is independent of the motion of the particle under 
consideration, and that this field has on an average no 
directional properties, shows that the mean values of the last 
two terms vanish, and the assumption that a stationary state 
* Communicated by the Physical Society : read March 13, 1908. 
