643 



that an infinite number of images lead to the same final-equation. 

 We may viz., as has already been shown by v. Smoluchovvski ^ 

 come to an integral-equation for f{x,t) in the following manner: 



When the particle in the time t has covered a distance between 

 a and a -\- da, this may have taken place in such a way that -. 



in the time i^ a distance is covered between w^ and .i\ -|- dx^ 



>> }> 5 5 ^a )5 >5 5 5 5 5 J) "^' J 5 5 •'^2 |" ^"^J 



" ' 5 5 5 5 fn 5 5 5 5 5 5 5 5 5 5 '''>i , , tt\^ — }- (J.Vf^ 



in which 2u^=:t and JS.i-, lies between a and a -\- da. When we 

 suppose that the times U are not too small so that the number of 

 shocks in U is great enough to allow the use of probability-calculus, 

 the probability of this occurring is given by 



in which ƒ is the required function. Now this product, integrated 

 for all values of a;^ . . . .i'„ for which 2.v-, lies between a and a -f- da, 

 must give the probability, that the particle in the time t has covered 

 a distance between a and a -\- da, so /{a,t) da, in which this /" 

 will be the same as the "elementary" ƒ from the above product, 

 when only t, is not too small. 

 So we get: 



00 X 



1 • • • 1 /(-^'i'^) • • -/i^T^tntu) d.r^ . . . <lx„ . A =f{a,t) da. . . (11^) 



■ 00 — 00 



Here J is a factor of discontinuity that satisfies the conditions 



A = when 2 a-., <^ a or 2.c^ ^a-i^ da 

 A= 1 ,, a<C. ^'V-, <^a -\- da. 



When n:=2 (Il«) becomes: 



J 



f{x,t,)f{a- xj,)dx=f{a,t,-^t^) .... {I[i^) 



One easily sees that inversely (II'») follows from (11^) so that each 

 of these equations may be used as a basis for the following surveys. 



We will now show in different ways how we may deduce the 

 formula (1«) from the equations (11). 



I. Take in (II«) the times t^ . . . tn small then n is great '). 



*) Comp. M. V. Smoluchowski. Ueber BowN'sche Molecularbewegung etc. Ann. 

 der Physik. 48 p. 1103, 1915. 



2) When we want, that in the first member of (Ha) f {x, t) is the required 

 function, then tv must be larger than a certain very small time, which Einstein 

 has estimated at 10-7 sec. for particles of a radius of I fx. (Ann. d. Phys. 19.371, 1906). 



