657 



Y. F (o, t) X 



So it follows from flie observations of Brillouin that s =z 1.5 X 10~". 

 As the maximum value of the probability is 1, a can never be 

 larger than Xj i.e. for the particles used by Brillouin >{<^0.1. But 

 X, will never be so small that, when f =: 1 or a not too small 

 fraction, we cannot take x = oo, without appreciably changing the 

 result, so that for these cases the boundary condition i^(0,^) = 

 gives a serviceable result. Strictly speaking the boundary-condition 

 can however never be F {0,t) =: 0, because^ then x must be infinite, 

 which is impossible. 



The probability e will be connected with the mechanism of the 

 collision of a particle against the wall and to get any insight into this, 

 it is desirable to know e as a function of different circumstances. 

 Except the experiments of Brillouin, no other experimental data to 

 determine s, are known to me. 



§ 4. Though the boundary-condition in § 3 is deduced only for 

 ,a plane wall, we may undoubtedly also apply it to a curved wall 

 in the form : 



W — 

 D^r=xF (XXIV) 



or 



where the horizontal line denotes values at the wall and r is the 

 normal directed towards the liquid. 



This we may apply to the case of a fixed sphere with a radius 

 R, surrounded by a liquid in which at the time / = many particles 

 are dispersed homogeneously («„ per unit of volume). The solution 

 of this problem is namely used by v. Smoluchowski ') in his theory 

 of the coagulation. Where he however uses as boundary-condition 

 F=0, we will here take (XXIV) as limiting-condition. The number 

 of particles that sticks in a unit of time after some computations 

 proves to be : 



dut yJ{ 6ii,R „ „ 



— = ijr RD 1,, = 4-T RD n, — ' , {XX V) 



dt " xR + D 'aK,R-\-D ^ ' 



when we, like v. Smoluchow^skl only consider times that are large with 



R^ , 



respect to — . 



1) M. V. Smoluchowski. Phys. Zeitschr. 17, p. 594, 1916. 



