105 



For the change in velocity we get then at each ini|»acl according 

 to Rayleigh 



where q is the rehxiion of the masses of particles and molecules, 

 V the velocity of the molecules. Now the |)roblem treated l)y Kaylkigm 

 in this way may be connected directly with the theory of the function 

 of probability for the way in the Brownian motiofi. If we lake the 

 velocity marked as vector, the terminal point is lemoved ± (jv 

 after every shock. The terminal point of the vector consequently 

 executes a Brownian motion at least according to the scheme which 

 is often given of it (cf. e.g. Mrs. de Haas — Lorkntz' dissertation), it 

 is certainly remarkable how Lord Rayleigh had already so long 

 ago deduced these results, which came to the foreground only by 

 Smoluchowski's work, which opened so many new views. 



It may have its advantages now it has become apparent that 

 Einstein's formula is the right one to say something further on the 

 kinetic mechanism. Let us first direct our attention to a single shock 

 of a particle of a great mass with a particle of -a small one. If 

 the velocity for the first is before the shock ii, after the impact ii, 

 the velocity of the small particle v and the relation of the masses 



q, where we have g*^!, then we get for every impact: 



u = u (1 — q) r±z qv. 



If we assume then that again and again after a time t a collision 

 takes place, then we have 



u — u' q , q 



X XT 



for every impact. We can only make a differential equation of this 

 equation of differences by taking t infinitely small, if q is of the 

 same order infinitely small and then we get 



du 



dt 

 where F may be written for /ir. Thus we see hero by a (not 

 very strict approach to the limit) Einstein's equation arise as it 

 were. If now we do not go to the limit, but avail ourselves of the 

 following graphic representation, its meaning becomes even more 

 clearly visible. On one axis we measure out the time (and to make 

 things easier we take again ecpial intervals between the impacts), 

 on the other the velocity. Between tyvo collisions the velocity is then 

 constant, at an impact the velocity suddenly jumps to another value 

 and this jump consists in every case in two parts; one part pro- 

 portional to the velocity of the particle before the shock with which 



