10 



c. V. L. Charlier 



m 





A„, 







1 



9 



9 







2 



30 



39 



4.33 



3.33 



3 



75 



114 



2.92 



2.50 



4 



190 



304 



2.67 



2.53 



5 



630 



934 



3.07 



3.32 



6 



1949 



2883 



3.09 



3.10 



7 



8335 



11218 



3.89 



4.27 



8 



27241 



38459 



3.43 



3.27 



9 



165190 



203649 



5.30 



6.07 



The divergencies are partly due to an incorrect fotometric scale, partly to 

 the insufficiency of our assumptions, {H and D constant), as will be found in the 

 following. 



3. Simplest assumption on P. The simplest assumption regarding the 

 projected absolute velocity P is obtained by putting the absolute velocity (v) itself 

 equal to a constant. What is the resulting frequency function for P supposing all 



directions of the velocity to be equally probable? 

 In other words, what is the probability that the 

 projected velocity will fall within the limits P 

 and P-]-dP? 



Evidently this probability is proportional to 

 the variation [dQ) in the angle 9, when P increases 

 from P to P + dP. Denoting with C a certain con- 

 stant the probability sought is hence Cd.Q. 

 But 



Fig. 1. P = t;sine •.• (^P=v cos 6(^6 



dd 



dd = 



dP 



dP 



'v cos 6 p2 



and the probability, before called dPf^{P), is 



CdP 



äPfÅP)- 



Vv^—P^' 



as long as P^v, whereas, evidently, cp^(P) must vanish when P > v. Hence 

 from (b) in § 1 we have 



vIp 



(8) 



a(p) = oiC dr B(r) 



giving the number dp a[p) of stars having a proper motion p ± ^/i dp. Putting 



y =zpr •.• pdr — dy 



