648 



is in first approximation an even function 7^ of the displacement 

 p — a^) . because the influence of the external force for small times 

 is small with regard to that of the Brovvnian movement. To take 

 into consideration the influence of the force, we determine the 

 displacement d, that the particle, starting from p, would undergo, 

 when there was no Brownian movement. So, tlie displacement in 

 consequence of the Brownian movement in the time /, is /; -|- d — a, 

 when we suppose that the two displacements are simply superposed, 

 /, being small. 



So for small t, we may substitute f/{p -\- d — a, t^) for f{p, a, t,) 

 whereby ff is a function that is even in its first argument. As the 

 influence of the force is entirely included by the quantity d, (p, 

 which relates to the undisturbed Brownian movement, has the 

 following properties : 



L{^,t)dè = l SiUÓ J^'cf (l,t)d§ = 2Bf. . . . (VI) 



To compute d, we make use of the fact that small particles, 

 under the influence of a constant force, cover a distance that is 

 proportional to the time, when we leave the influence of the 

 Brownian movement out of account. As however the force rp(x) is 

 a function of x, the distance d, which has been covered, is a function 

 of t, that will also contain the higher powers of /,, being expanded 

 into a series of ascending powers of this quantity. As t, is chosen 

 small, so that in every instance we keep only the first power, we 

 may take for d: 



d=^t,xp{p) (VII) 



So (V) becomes : 



X 



f(w, p, t,) if \p~a + ^,^ (p), t,\ dp =f(x, «, t, + <,). 

 By taking p z=z a ^ 'è we may expand ƒ and q as follows: 



1) Though we do not want the form of this function, we may remark that 

 this will be: 



1 4Dt.-. 



(f (p — ■ a, tA = " 7 - e 



