432 MATHEMATICAL APPENDICES 38* 



Consequently the mean value which we obtain from this other 

 consideration is the value already obtained, 



L = 0/r, 



This mean value L is greater than the other M(l], since, in the 

 summation of the B free paths I of any particle, the larger values 

 I of the faster particles come more into account than when only 

 one path for each particle is considered, as B too increases with 

 the speed w. 



We must give up the idea of calculating with exactness the mean 

 value M(l), by reason of the complicated form of the function B. 

 But we can obtain an approximate value for M(l) by a tolerably 

 simple calculation if we substitute for B its approximate value 



as given in 32*. If, then, we put 



the integral takes the form 



Mil) = 4V 2 7r 

 which reduces to 



==4v/27r- ] 



L V," ~J Jw- 



and from this we may calculate the mean value. By the help of 

 tables l of this integral we find 



M (I) = 0-937 D. 



This value is certainly less than L, but we must still remember 

 that it is only approximate. For we have put too large a value 

 for B, and have consequently got too small a value for M(l). For 

 the values of w, which occur the most frequently, the error in 

 the approximation to B is about 2J- per cent., and thus the factor 

 G'937 is too small by this amount. The mean value M(l) is there- 

 fore about 4 per cent, smaller than L. 



39*. Interval between Two Collisions 



In calculating the average interval between two successive 

 collisions of a particle with others, we can arrive at two different 



1 Bess el, Fundamenta Astronomies, 1818, p. 36. En eke, Astronomisches 

 Jahrbuchfilr 1834. 



