studies in Stellar Statistics 



27 



we obtain indeed 



(58**) M„lp) = M[P) [år CO D{r) r" '£^[m — 5 log r) X ^ : [ dr a> Z;(r) cpo(w — 5 log r) . 



The quotient between the integrals is, however, nothing but the mean value 

 of 1/r. Hence the important result (introducing tu = 1/r). 



(58**) M^p) = M,iP) . M^iiz) 



or 



(58***) M.^{-n) = M,n{p):M,iP)- 



or ill words: the mean value of the parallax of the stars having the magnitude m is 

 equal to the mean value of the proper motions of the same stars multiplied, by a con- 

 stant factor*) (= 1/M(P)). 



In this manner we are able to obtain, from the proper motions, a value of 

 the. mean parallax of the stars of different magnitudes and thence to solve the 

 fundamental problem in stellar statistics without knowing the function 'fo(ilf) which, 

 however, can now be indirectly determined — as well as D(r), from a{m) and M,„(^). 



The formula (58***) can be derived directly from the relation 



(58****) p = P:r 



if we take in consideration that the frequency function of P (and hence also that 

 of the absolute velocities) is supposed to be indépendant of r. In a more thorough 

 treatment of the problem the relation between P and r — which without doubt 

 exists — must be taken into consideration. This cannot, however, be done before 

 the motions of the stars have been more completely investigated. 



We are now able to sketch the outlines of the method to be followed for 

 obtaining a first approximation to a knowledge of the structure of the Milky Way. 

 This will be done in my next lecture. 



13. Putting 



r = e""-' {b = 0.2/mod.) 

 and using the formulée (21) and (21*) our formulée are now 



+ 00 



(59) a{m) =fdy à^{y) (po(m -f , 



— 00 



(59*) a{m) M^p) = M[P)[dy ^,[y) rpo(m+^) e'' , 



— 00 



where 



(59**) ^fyy) ^lübl) {e~^^) e"^^" = o^b D[r) r\ 



*) Obs.! This factor may differ (owing to the motion of the sun and other reasons) from a 

 square to another. 



