The motion of the stars • 99 



the mean velocity of a particle is inversely proportional to the square root of the mass 

 of the particle. 



Is it allowed to extend this conclusion to the motion of the stars? 

 Let us examine the consequences of such an assumption. 



Let ']t(u)dH denote the number of stars, of the mass (jl, having velocity com-' 

 ponents, in a certain direction, between — ^ du and u -\- ^ du. Then according to 

 the law of Maxwell. 



(97, ,H„)=J£L;-'. 



a J/ 2n: 



where a denotes a certain parameter, constant for all stars. 



In a population of stars, say all stars down to the sixth magnitude, there is, 

 certainly, a great variation regarding the masses of the stars. Most simply we 

 may assume that the frequency curve of the logarithmus of the masses is a normal 

 curve of type A. 



Putting 



(98) [JL = ■.■ ^ = nat. log [1 



we suppose f[z)dz to denote the number of stars for which the logarithmus of the 

 masses lies between the limits 2 — \ dz and 2 -\- \ dz, where 



(98*) f[z)^. — —e , 



L being the dispersion in z. 



Let further ^''[u)du denote the total number of stars — of all possible masses 

 — with velocity components between u ± \ du then 



+ 00 



(98**) W[u)=jdsf{3)'^{u). 



— 00 



The direct evaluation of this integral is not possible, but the form of the 

 function W[u) can, nevertheless, be easily found from the moments, which quantities 

 may be obtained from the integral above. 



Let ^2,. be the moment of the 2r"' order of ^(w), so that 



+ <» 



n^^ = j duu^ ^(w), 



and observing that 



+ 00 2)- 



\ duu'^{u) = 1. 3. 5 . . . 2r— 1 ^ 

 = 1. 3. 5 . . . 2r— 1 



+"> 



L 3. 5 . . . 2r— 1 a"Y dzf\z)e~"\ 



we get 



2r 



