649 



We also expand the second member of (V) : 



df(x, a, i.) 

 ƒ (.^^ a, t, + Q =f{.v, a, t,) + t, -^-— ^ + • • • 



After having multiplied and performed the integrations, we find, 

 making use of (VI), when we keep only the terms with t.^ and drop 

 those with higher powers : 



O^j Oa^ da da 



or 



^ = .ÖT^-/^-(^f!/) (1^^^^) 



di Oa^ Öa 



This is the generalised equation of diffusion that v. Smoixichowski 

 uses, however, deduced from the integral-equation (V). 



When one does not restrict oneself to the problem of one dimension, 

 but investigates the case of three dimensions, the probability that 

 a particle with the coordinates x, y, z will have after the time t 

 coordinates between />, q, r and p -|- dp, q -\- dq, r -{- dr may be 

 represented by f {x, y, z, p, q, r, t) dp dq dr. 



We easily understand, that this function satisfies the following 

 integral-equation : 



OO CO 00 



IIS' 



f {x,y^z ,p,q,r,t^) f {p,q,r,a,b,c,t^) dp dq dr=f{x,y,z,a,b,c,t^-{-t^) . . {IX) 



X X — CO 



When we call again the force, that is now a vector, \p, we find 

 for / in an analogous manner as in the preceding, the equation of 

 diffusion : 



^J. = nLf-l3div(x^f) (X) 



The operations A and div. here refer to the second three inde- 

 pendent variables in ƒ, on which tp also depends. 



The truth of (A^) is also easily proved in another way. 



With the help of (VIII) we can solve the following problem: 

 The plane .^=0 is formed by a fixed wall, which, on a particle 

 in the liquid by positive .v, exercises a repelling force, which becomes 

 perceptible in a point ^ at a small distance of the wall, increases 

 by approaching the wall, and becomes infinite when x = 0, remaining 

 so for negative x^). What is the probability that a particle for which 



^) A wall with the described properties will reflect tlie particles colliding against 

 it, without ever one of them adhering. 



