26 



C. V. L. Charlier 



and that the secular difference, between [x and m is eliminated, then the true number 

 of stars having the magnitude m ± dm is 



-^e~ dm 



where 



(56**) k' = k^^ — a'. 



12. Our fundamental equation I. can easily be extended to an}^ number of 

 characters that are simultaneously taken into consideration. Take two characters 

 i?, aud J?2 suppose 



E,)dE^, dE^ 



to be the probability that simtdtaneonsly and E^ lies between the limits E^ ± V2 dE^ 

 and ± '/s dE,^, let further e^ and e^ be the apparent intensities of these characters, 

 then the number of stars having the characters considered in the intensities 

 ßj ± V2 de^ and ± 7^ de^ is given by «(e^ , e^) de^ de^ where 



00 



(57) a[e„ = co I dr B[v) ^ 'f(^\, E,). 



0 



With this formula we will solve the following 



Problem: What is the mean value of the proper motion of the stars having the 

 magnitude m ± V2 dm ? 



Suppose the absolute velocities of the star to be independent of the distance 

 r from the sun and let 



^q{M) be the frequency function of M 



cpj^(P) » » » of P (= projected absolute motion) 



and put 



ïE'" = (0 D[r) r^ (p,, {ni — 5 log r) 9 ^(pr) . r 



so that W dr dm dp is the number of stars (in o)) in the distance r having the 

 magnitude m and the proper. motion p, then the mean value of p is given by 



00 GD CC 00 



(58) M4p) = I jpWdrdp-.jjWdr dp. 



r=0 p=0 0 0 



Owing to the fact that P is independent of r this double integral can be 

 transformed into the product of two single integrals. Putting 



CO CO 



(58*) M{P) =^jdP P'£^{P) : / dP cp,(P) 



0 0 



t 



