1269 



'1 



When we further bear in mind that A„^=z [/bt and v,n^= y ., 



when A„, represents the mean displacement and v„i the mean value 

 of the time average of the measurable velocity, we see that y,„ decreases 

 with increase of t. It is therefore natural to suppose that a measurable 

 velocity can be introduced that decreases with the time, thus being 

 smaller in the later sub-intervals than in the first. If in virtue of 

 this the force of friction at the end of t should also be put smaller 

 than we did, namely at half the value'), we should iind for A' 

 exactly the value of Einstein's formula. 



A simple calculation, however, teaches that the desired correction 

 is not to be obtained on the gi-ouud of these considerations. For 

 this we point out that the chance that a particle gets a deviation 

 A in a time t, is represented by : 



C e '"" ^A in which m' = f A~\ 



When we now choose the group of particles that all have the 



same A, and when we divide t into two sub-intervals t^ and t^, 



the different particles of the A-group will travel over different 



paths A^ in the time t^, and over different paths A, in the time i^, in 



'which of course A^ -|- ^j = ^ = constant for the group. 



When for this group we now examine the middle value of Aj, 

 a simple calculation teaches: 



— A t 

 A, =A-i 



t 



— ^ t 

 so that also : A, =:x A - 



• 



In this it has been assumed that the values of Aj and A, are 



statistically independent of each other, when not a A-gi-oup, but the 



collection of all the particles is ('onsidered. It is known that this 



may be assumed as long as /, and /, are not too small, i.e. /, and 



tj must be sufficiently large to allow us to neglect the influence of 



the initial velocity and of the initial force. 



It appears from this that we may divide the velocity of the 



particles during the interval t into two terms: a nnifonn velocity 



A 



— and an irregular term, which, iudepeiideut of the value of A for 



1) We should exactly obtain this factor '/j when we did not derive Vm by 

 dividing [\m by t, but by ditTerentiating Aw with respect lo t. 



