studies in Stellar Statistics 16 



(20) Mj,{r) ßr D{r) 'f pr) : [dr D{r) >•» f^{pr) 



0 'o 



(20*) , =C,:p. 



and 



ilfp,(r) : Mp,{r) = p,:p^ 



6. Reduction of the equations for h, p and 5 to the same form. 



Putting in (11*) 



b = 0.2 mod. = + 0.46064 ; p = — by 



we get 



+ 00 



(21) =fdy (p^(ni + p) 



— 00 



where 



(21*) AJ^/) = CO ft Z)(e- 



and the corresponding expression for the mean distance 



+ 00 



(22) aim) MJjr) =^ j dy f^(m + y)e ' 



■by 



— GO 



Passing on to the proper motion we observe that log h and log r have been used 

 conveniently as variables in the study of the brightness and we shall find that logp 

 and log S are proper variables for the proper motion and for the distances of 

 double stars (in stead of p and S). 



Putting 



(23) P=e'; p = e^; 



we may find tlie expression for the number of stars (a(C) dCj for which nat. log p 

 lies between the limits C ± Vs d'C,. Let 4>i(Z) be the frequency function of Z and 

 using the general formula I we obtain immediately 



00 



a(C) = (üjdr D{r) r= 4>i (C + nat. log r) 



or putting r = e? , 



(24) a(C)=/(^pA(p)<ï>j(C + p), 



+ 00 



where 



(24*) A(p) = w D{e^) e^P 



The expression for the mean distance of stars with the proper motion p ± V» dp, 

 or more precisely with nat, logp — 'C, ± ^It d^, is 



