1267 



{)aiti('l^s which all have the same w{t^) on t^, and when we take 

 the iu{^) for these particles at a later moment, and then the mean 



of 'iu{^) over the group, which quantity I sliall represent bj iu{^) 

 we shall find that the particles have assumed all kinds of values 



of tu, so that iu{&) *=0. At least this will be so when (^ — ^J is 

 not very small; only for very small values of [^ — t^} there exists 

 correlation between the values of w{&) and iu{ti), and then the 

 product w{^) iv{t^) will be positive on an average. 



This would, indeed, be accurate, and would lead to the opposite 



sign for (5c), when tui^) ' approached to zero aperiodically. This 

 is, however, not the case. In order to see this we observe that in 

 virtue of the mutual independence of distribution in configuration and 

 in velocity we may write in a notation that is easy to understand : 



^v,, = 



J' 



When we take: 



t 



h 

 and when we again average for the w(t^) group, in which it may 

 be assumed that with sufKiciently long t — t^ the initial w[t^) has no 



-• — H'('l) 



longer any influence on the final velocity, so that x(0 =0, we find: 



■ R'Cl) 



\io{0)dO =0 (11) 



zwiU 



When w{t^) is thought positive, w{d) ' will also be <^ for 

 very small values of 0. The fact that the integral is zero means there- 

 fore that the positive interval is succeeded by a negative interval, before 



the value of iu{6f ^ falls to zero ^). When under the integral sign 



^) This may also be expressed by statihg that w[6) w(d+S) is positive for very 



small values of 5; for somewhat larger values it is negative, descending to zero 



for large values. This change of sign of the product has been overlooked by 



Ornstein (Zitlingsverslag Dec. 1917, p. 1Ü0S, § 2). In consequence of this he arrives 



d - 

 at the remarkable conclusion that the assumption j u^ = is not justified, b'or 



according to his compulation it follows from this that u* is not constant, but the 

 sum of a linear and a periodic function of i\ 



