o 



ƒ 



927 



{nt 3111 nt f cos nt — 1) dn ^). 



o 



The sign of tliis integral may for larger values of t be made 

 quite arbitrarily by proper choice of f{n). That it should be essen- 

 tially negative is consequently not true*). 



3. In the quoted paper by Ornstein the first theory of the Brovvnian 

 motion as developed by Dr. Snkthlage and J. D. v. n. Waals was 

 criticised on the basis of the fact that if comes into conflict with 

 the theorem of eqnipartion. 



There the thesis was made use of that 



\ L{h)sinQ{t-g)dS,^^ . ...... (5) 







is proportional to t. Here iv{^) is a function subjected to chance, so 



that the average value is zero"). In a note van der Waals says: 



"This change of sign (of iü{6) iv{^-\-ö) was overlooked by Ornstein. 



In consequence of this he ai'rived at the remarkable conclusion, that 



d — 

 it is not allowed to accept that -- ii* =:0. For from this it follows 



dt 



according to his calculation that u" is not constant, but the sum of 

 a lineary and periodical function of t\" 



It is necessary to remark in contradiction to this, that the diffe- 

 rential equation*) of v. d. Waals — Snethlage viz. 



') For ^ = 00 this expression becomes equal to — /"(O), is thus essentially positive 



(i.e. /" is essentially positive). 



For very great values of t we can require that the average is cc {t), then we get 



/(n) 2 fr/(A) . 



— sin n/ a/ 









for very small values of t the average value is also positive. 



■} The proof thai v. d. Waals gives of the disputed thesis by differentiating 

 A^ (p. 1331 of his paper) is not right. The formula L^ = bt is deduced by a 

 transition to a limit, and the process is such that in differentiating l^ we do not 

 get b as L cannot be differentiated. 



3) Compare Ornstein, these reports XXI, p. 96. 



^) The equation, which is treated by both authors as a differential equation, 

 does not apply, as they suppose, to arbitrary kinds, but only to the commencement 

 of the movement, compare Ornstein and Zkrnike, These Proc. XXI, p. 109. 



60* 



