647 



When one chooses however another image, i.e. another fornr for 

 the "elementary" function /{,i\), one gets the same formula {h), 

 but then D depends on the choice of ƒ (^v) in another way. ^) Only 

 when the "elementary" function is the required /{j.\,Q, D will be 

 an indetinite constant, which is determined by the nature of the 

 Brownian movement. 



§ 2. In the preceding paragraph we have always supposed that 

 the Brownian movement is not influenced by an external force. In 

 consequence of this, we might, for reasons of symmetry, accept 

 fU',t) to be even in ,r. When however a force acts upon the particles, 

 the probability of a certain displacement will be no longer an even 

 function of this displacement. Nor will this probability only depend 

 on the displacement, but also on the starting-point. First we will 

 consider displacements in the direction of A' only. When we call 

 the abcissa of the particle at the time t = .v, we may represent 

 the probability that this abcissa at the time i lies between /> and 

 p -\- dp, by f{x,p,t) dp. One easily understands, that this function 

 f also satisfies an integral-equation viz. : 



ƒ 



f{x,p,t,)f{p,a,t^)dp=f{x,a,t^^Q'). . . . (F) 



From this integral-equation v. Smoluchowsk[ ') for the case of a 

 quasi-elastic force, has determined ƒ by repeated integration, but 

 hereby he remarks, that this way is only serviceable when we make 

 a simple supposition about the force ip, and that, in general it is 

 preferable to determine ƒ from a generalised equation of diffusion. 

 This equation is deduced by v. S. on a somewhat phenomenological 

 manner, without making use of (V). We will now show how this 

 equation may be deduced from (V), when we use suppositions that 

 are also put forward by v. S., determining ƒ by repeated integration. 



When namely in (V) we give t^ a very small value, f{p,a,t^) 



1) Suppose e.g. that regularly after the time t the displacement I may occur 

 to the right or to the left, each with the probability — ^- , so that the chance 

 of remaining at rest is ^•. By this the form of f {Xy,) is defined and we find for D: 



r 2xJ 



2r 



3) See also M. Smoluchowski Ann. d. Phys. 48 p. 1103, 1915. 

 ■^) Ann. d. Phys. 48 p. 11Ü4, 1915. 



47* 



