646 



When we consider that 



00 CO 



Cf{x,t) (Li: = 1 and \x^'^+^ / {.v,f) dx = . 



When in this we take ti small enough to allow the neglecting 

 of following terms and further 



CO 



we get the equation, 



h-s - ■"-I 



df d*f 



As here — and -— are functions of a and if,, D will not contain 



d t da* 



t^ and therefore will be a constant. ^) 



It is remarkable that we find here the diffusion-equation as a direct 

 consequence of the iMtegral-equatioii (H''), while (IV') immediately 

 connects the diffusion-coefficient with the mean displacement. 



This equation (IV') has to be solved without boundary -conditions 

 and with the beginningconditions : 



00 



da ^=\ , 2*. ƒ {a, o) = except for ^ = 0. 



This is a well-known problem of diffusion or conduction of heat, 

 the solution of which is given by (1=^). 



The exponential function of(I'^) satisfies the integralequation (11**) 

 for arbitrary n and t^. When we take however for the ''elementary" 

 function /(^v) in the first member an arbitrary function, that only 

 answers to the conditions ' 



1 ƒ (a*v) is even, 



2 /(.^v) differs perceptibly from zero only when Av has a small 

 vabie, 



the integral in the first member of (11=*) will, for great n always 

 give the same f (a, t). ") Therefore we understand that the simple 

 image of the Brownian movement, that is given by v. SmoLuchowski 

 yields a right result, in spite of the fact that it approximates the 

 true movement very badly. 



1) When we introduce the solution of (IVb) i. e. (la) in (IVa), D becomes 

 actually a constant, as is supposed in the solution of (Wh). 

 ~) See note 1, p, 645. 



