106 



the velocity decreases and one part wliicli may be either positive 

 or negative (and in general may posses all sorts of valnes dependent 

 upon the conditions of the impacts, which in the simple case investigated 

 by Raylkigh is ± qv). The velocity-time curve is thus a discontinuous 

 curve. If the velocity has become large it has the tendency to 

 become smaller by shocks owing to the first part, whilst the second 

 part exercises no systematic intlnence in a conti-ary sense. If now 

 we imagine a combination of curves drawn starting from a given 

 velocity, Einstein's equation will represent for each of these dis- 

 continuous curves the differential equation. At the same time if we 

 introduce the curve u = u^e~^' into the scheme, this line will at all 

 times be an average of the discontinuous velocity time-curves in the 

 diagram. 



§ 4. Finally I will deduce the function of piobability for the 

 Brownian motion under the influence of an external force. We 

 take this force km, where k depends upon the place (.s-). 



The equation of motion for our particle is then the following 



— =-/?« + F -f /t (19) 



dt 



If now a particle has in the time ^ = a velocity ii\ if in a 

 time t — T the velocity has become u' and a way s' is accomplished, 

 and if n and s represent these magnitudes in a time t, we get 



t t 



u — u ^ \ '] ', 



t—z t—r 



— /J(s— s')+ IFdt^ Ikdt. 



We now consider the time so small that the way accomplished 

 in that time is small enough to treat K in the last integral which 

 depends upon .s* as a constant. 



We have thus 



u—u'z=-i3(s~s')+iFdt^-kT (20) 



Now we want hs = s — s' and As\ In order to determine these 

 we apply (3), this yields 



u—u' 1^ M„ <?~''' (1 — e+i^') 

 as we have to take the mean value of A^ for all possible values 

 of w„ the average of m.9 being zero we get 



^'Ks = kr ........ (21) 



