634 REPORT— 1900. 



case eacli value of the resultant velocity would occur witb a frequency propor- 

 tional to its square, and a factor such as c'^^ is required to keep down very high 

 values. The generalisations by Boltzmann and Maxwell to internal degrees of 

 freedom would lead us too far, the aim here proposed being merely concrete 

 illustration of the very general but purely analytical argument that is fully set 

 forth in tlie treatises of Watson, Burbury, and Boltzmann. 



2. The Partition of Energy. By G. H. Bryan, Sc.D., F.R.S.^ 



Consider a system of particles in a field of force acting on one another with 

 forces which are functions of the distances between them. If u, v, w are the 

 velocity components of a particle of mass m, V, the potential energy of the system, 

 the rate of increase of the component of kinetic energy, ^ mu", is given by 



d ,, „. dV 



— (i mu-) =^ ~ u — 



If the probability of any given motion of the system is equal to tlie probability ot 

 the reversed motion for given positions of the particles, then since equal positive 

 and negative values of ti are equally probable it appears that the mean rate of 

 increase of i mu- estimated from probability considerations is zero. Now form the 

 second differential coefficient of i 7)iu~ with respect to the time, which may be called 

 the acceleration of this energy component. AVe obtain 



~-U mu-) = - \ -u2[u~ + v-- + w^ 1— _ 



dl.-^- m\dx) \ d.L- dij dzj dic 



If we are given the probability that the coordinates of the system may be between 

 given limits, then a condition for the stationary state is that the mean values of 

 the accelerations of ^ mir, ^ }izv'-, i mic- are zero. We thus obtain a system of 

 equations of energy equilibrium for the system, which are sullicient to determine 

 the mean values of the components of kinetic energy, provided the system is such 

 that the mean values of products of velocities such as MjVj, ?/jMo, or u^v.^ vanish. 

 If this is not the case the conditions for a stationary state involve writing down 

 further expressions for the accelerations or second differential coefficients of these 

 Telocity products, and equating their mean values to zero. 



In this way the mean values of the squares and products of the velocity 

 components for a stationary distribution are expressible in terms of the mean values 

 of the squares of the force components, and the second differential coefficients of the 

 potential energy with respect to the coordinates. 



In this preliminary investigation the .simplest possible illustrative examples 

 are considered. For a system of two particles moving in a straight line and acting 

 on one another with finite forces, the partition of energy follows Maxwell's law, 

 and the mean product of the velocities vanishes if there is no external field of 

 force. If there is a field of external force, this is no longer in general the case. 

 We thus have some justification for the belief that in a polyatomic gas Maxwell's 

 law of partition may no longer hold good, and this may account for the experi- 

 mental result that this law is verified approximately only when translational and 

 rotational energy are alone taken into account. 



The principal advantage of studying the problem of energy-partition from the 

 consideration of energy accelerations is that it leads to results for a perfectly 

 reversible dynamical system somewhat analogous to the irreversible properties of 

 temperature. The property that heat tends to flow from a hotter to a colder body 

 is represented on this view by the property that when two stationary systems are 

 allowed to act on one another, then if a certain inequality is satisfied energy is 

 accelerated from one system to the other, and the direction of the acceleration is 



' This paper wil) be published in extenso in the dedicatory volume to Professor 

 Lorentz published by the University of Leiden. 



