1908-9.] On Energy Accelerations and Partition of Energy. 349 
XX. — On Energy Accelerations and Partition of Energy. 
By C. W. Follett. Communicated by Professor W. Peddie. 
(MS. received June 17, 1908. Read July 6, 1908. Revised January 27, 1909.) 
(1) In the volume of the Arch. Neerlandaises for 1900, Dr Bryan discussed 
the investigation of the distribution of the velocities of a dynamical system 
for a stationary state from probability considerations. 
He suggested the method of obtaining expressions for the second 
differential coefficients with respect to the time of squares and products 
of velocities such as enter into the expression for the kinetic energy, and 
finding from them equations of energy equilibrium, as well as conditions 
of stability for the stationary state. 
He applied the method to a few simple examples, and indicated in these 
the preponderance of conditions for non-equipartition of energy. 
Dr Peddie, in two papers ( Proc . Roy. Soc. Edin., vol. xxvi., 1905-6, 
and vol. xxvii., 1906-7), arrived at similar conclusions by an entirely 
different method. Dr Bryan in his paper advanced the view that the 
conditions of stability may possibly afford the true clue to the partition of 
energy problem, while the limitations required may explain the failure of 
Maxwell’s Law of Equipartition to account for many physical phenomena. 
The present paper contains an extension of Dr Bryan’s work to the 
case of two particles moving in a field of force and under the influence 
of their own attraction (or repulsion), and applications of the results 
obtained to certain illustrative examples. It is necessary to make certain 
hypotheses which will be defined in the course of the work, as the com- 
plications which ensue remove the possibility of treating the problem in 
its complete generality ; the assumptions which will be made cannot, for 
instance, be justified in the case of a Newtonian field of force. 
The method adopted should have an important application to the theory 
of gases especially ; the two particles may, for instance, be considered to 
form a di-atomic gas-molecule. 
We shall use the term molecule to denote the system of two particles 
under consideration ; the two particles themselves may be defined as 
atoms. 
(2) The probability that the co-ordinates of the system shall lie between 
assigned limits is supposed to be given, and is of course a function of the 
