282 G. H. BRYAN. 



The left hancl side dénotes what we sliall call tlie mean accélération 

 of ilie h'rneilc eneriju of tlie pai'ticle. The significance of the meau values 

 will be more easily realised if we thiiik of a mimher of particles distri- 

 buted according to a statiouary law^ the probability of giveu coordiiiates 

 and velocities being propoTtional to the iiwnher of particles haviug those 

 coordinates & velocities. 



The conditions for a stationary distribution require that tlie mean 

 accélération of energy shall be zéro just the same as conditions of equi- 

 librium in statics require that the accélérations of the bodies of the 

 System shall be zéro. The équations re]3resenting thèse conditions we 

 shall call erpiaflous of eiiergij equilibr'mm. Eor the présent case the équa- 

 tion of energy equilibrium gives 



[^-] 





We see at once that energy-equilibrium is impossible unless the mean 

 value of d~J'''jfiy~ is positive. But even if the particle is moving in a 

 région in which d-Jjdx^ is everywhere négative tins condition raay still 

 be broiight about by supposing the région bounded by perfectly elas- 

 tic Avalls the etfect of the forces called into play during impact being to 

 increase the mean value of d~ ]'\dx~ by a tinite amount. 



The condition of stability of energy equilibrium is obtained by 

 putting 



[i;..^] 



T, -' 



r 



where 7',, is the mean kinetic energy determined by the équation of 

 energy e(|uilibrium^ and s is a small variation in the kinetic energy which 

 may be due to initial disturbance. We thus obtain 



dfi ~ m L dj'^ J ' 



For stability the variations in s must be periodic, and this condition 

 will be satisfied if [d^FIdx-] is positive. Thus the condition of energy 

 equilibrium in volves in this instance the condition of stability. If the 



