ENERGY ACCELERATIONS, A STUDY IN ENERGY, &C. 281 



te in is equally probable with the reverse motion. In this paper it is pro- 

 posée! to apply tlie raethod to a few of the siraplest possible illustrative 

 examples, leaving a more gênerai solution of the problem for future inves- 

 tigation. The présent exemples will sufïiciently indicate the characte- 

 ristic features of the results to whicli the method leads. 



Example I. lif'diUuear inolloii of a single partit-lK. Consider a par- 

 ticle of mass w. moving along the axis of .;■ iu a field of force, the poten- 

 tial energy of tlie particle due to the field being /' a given f nnction of 

 ,/•. Then the rate of increase of its kinetic euergy is given by 



à /-X .A (h ^, dV 



dt U """) = '"" Jt = "^ = "'ih- 



wliere A is the force on the particle. 



Suppose now that the actual motion of the ])article is unknovvn and 

 we require to find out wluit we caii about the probable motion from pro- 

 bability considérations. Let ihe assumption be made that for any given 

 position of the particle the probability of its moving with a given velo- 

 city is equal to the probability of its moving with the reverse velocity. 

 In other words the probability of ." lying between q and (i-\-dq is assu- 

 med to be equal to the probability of its lying between — q and — q- — dq. 

 Then it is clear that the estimated average rate of increase of kine- 

 tic energy is zéro since the probabilities of this rate of increase having 

 equal ])ositive and négative values are equal. Now let us form the second 

 dirt'erential coefficient of ^ mo" We get 



d- /\ ,\ ^^do , dX X2 , dXdx 



_ J^ , dX _ l /d^\r _ 2 ^' 

 m dx m \ dx J dx"^ 



Now let f{x)dx be the probability that the coordinate may lie be- 

 tween X and x-\-dx, and (p {o) do the probability that the velocity may 

 lie between v and v-\-do. Multiply the last équation hj f{x)dx(p{v)dr, 

 and integrate. Then using square brackets to dénote mean values 

 we hâve 



