Btudies in Stellar Htatiitics 



11 



we obtain 



where is a certain constant. 



rix 



VT- 



The probability ol: a value P ± V2 rfP (assuming v — const.) was 



Integrating between and P^ we obtain the probability for finding a value 

 of P between these limits to be 



G 



arcsm 



arcsm 



V V 



As P must necessarily be between 0 and v we must have C = 2 /k (or Vso, if 

 the arcsines are expressed in degrees). The following little table gives a specimen 

 of this formula 



Pjv 



arcsin P/v 



A arcsin P/v 



'/90 A arcsin P/v 



0.0 



O'OO 







0.1 



5.74 



5.74 



0.064 



0.2 



11.53 



5.79 



0.064 



0.3 



17.45 



5.91 



0.066 



0,4 



23.58 



6.13 



0.068 



0.5 



30.00 



6.42 



0.071 



0.6 



36.87 



6.87 



0.078 



0.7 



44.43 



7.56 



0.084 



0.8 



53.13 



8.70 



0.0971 



0.9 



64.16 



11.03 



0.123 \ 0.507 



1.0 



90.00 



25.84 



0.287) 









1.002 



The numbers in the last column give the probability of a value of P within 

 the limits 0 to O.l v, 0,1 v to O.2 v a. s. f. We find that large values of P are more 

 probable than small ones. More than a half of all values are greater than 0.7 v *). 



Integrating the value of dp a(p) betwen p = ao and p, we get the number 

 — A{p) — of stars having a propre motion > p. This number is 



(9) ^(f)=i- 



Actually it is very difficult (or impossible) to obtain through observations the 

 true number of stars having a proper motion greater than, say, 0[i. The problem 

 must, indeed, be treated somewhat differently, as will be seen beneath. Never- 

 theless it is interesting to compare the formula (9) with the observational results, 



For the Bradley stars it is found (proper motion = p.) 



*) The mean value of P is -v = 0.6366 v. 



It 



