14 C. V. L. Charlier 



The number of stars having the proper motion jp ± dp is a[p) dp, where 



(16) a(p) — (üj dr D(r) ^^ipr) 



o 



or substituting the value (12) of D{r) 



(16*) a{p)= C :p^-\ 



Integrating between p = oo to p we obtain for the number A{p) of stars with 

 a proper motion > p 



3 — s 



(17) Aip)=C,:p 



Proceeding to the mean distances we first shall deduce the mean distance 

 of a star of the magnitude m± '^l^dm. From (10) und (II) the number of stars at 

 the distance r ± '/2 dr having the magnitude m ± ^/a dm is 



(18) dm dr co B[r) «^(f/« — 5 log r) 



When determining the mean distance for stars of the magnitude w, we have 

 to consider m as a constant. Hence we get 



(18*) Mm{r) — jdr B{r) r^ cpo(m — 5 logr) : jdr D{r) ^^(m — 5 log r). 



0 0 



Substituting m — 5 log r = x 



we obtain, after some reductions (using the value (12) of D) 



(19) Mm{r) = Ke'"', where i = + 0.46052 

 which gives 



(19*) ilfm + i(r) : M^r) = = antilog 0.2 = 1.B85 = Jf^^r) : M,n + i{Tt) 



(if parallaxes are used instead of the distances). The following httle table gives the 

 values of the parallax for stars of different magnitudes according to (19*). 



m 



lf(7t) 



(Kapteyn) 



1 



o."ioo 



0."060 



2 



0.063 



0.044 



3 



0.040 



0.033 



4 



(0.025) 



0.025 



5 



0.016 



0.019 



6 



0.010 



0.014 



7 



0.0063 



0.010 



8 



0.0040 



0.008 



The constant K in (19) is here so determinpd thet we get for m = 4"'.0 the 

 same value (0,025) as Kaptetn. 



For the mean distance of stars with the proper motion p we obtain in like 

 manner (supposing D = yr" '') 



