617 



and retarding ones that tliebe different i-esults must be ascribed. 

 This is evident from considerations of von Smot-uchovvski, who, thongh 

 assnniing such a resistance proportional to v, obtained the second 

 residts by simply taking tiiis resistance propoitial to a/' and (,>. 



The reason of the difference lies in the assumptions on the sliding 

 at the surface. Without sliding the resistance is found proportional 

 to (I, and not to a* and <>. As long as the velocities of the mole- 

 cules may be considered as independent of liiose of the particle, 

 the assumption W proportional to Qa'v is the more natural one. 



The number of collisions is then proportional to the surface of 

 the particle and the density, while the total force is also proportional 

 to V by the added opposite influences of the forces on the fore- 

 and back-side, each of which is pro[)orti()nal to the square of the 

 relative velocity. This resistance has been calculated by Cunningham ^), 



who found : | a*vn 



{M-\-m)h 



When howevei- the velocity of the molecules depends on that of 

 the particle the relative velocity v can again depeud on a, so that the 

 dependency on a becomes different from that in the preceding formula. 



It is therefore evidently not true that the assumption of a resisting 

 force, represented by ^/; = />i; should be connected with the supposition 

 that there is no sliding. 



It is the last assumption that is of importance here. 



Van per Waals and Snkthlagk have objections to this assump- 

 tion. When the molecules of the fluid participate in the motion of the 

 particle, the velocities of the molecule are no longer independent 

 of the velocity of the particle, which should however be the case 

 in the here existing state of statistic equilibrium. This participation 

 of the motion would e.g. be possible in the case of the fall of a 

 particle under the influence of giavitation, but not here in that of 

 the Brownian movement. 



Here it may be remarked however, that Just by the resolution of 

 the forces into accelerating and retarding ones an analogous case has 

 been created. The accelerating forces take the place of gravitation. 

 The action of the molecules on the particle is resolved into an acci- 

 dental, irregular one, which may be regarded as the cause of the 

 motion and a regular, resisting one which represents the friction. 



When we consider <( font/ time there is statistical equilibrium. 

 There is however a continual transformation of energy of the 



') E. Cunningham: "On the Velocity of Steady Fall of Spherical Particles through 

 Fluid Medium". Proc. Roy. See. Ser. A. Vol. 83 p. 357. 1910. 



