studies in Stellar Statistics 7 

 Substituting for E its value in terms of c we have 



dE=—de 

 "de 



{r being a constant in the element of volume considered). 

 Hence the theorem: 



The number of stars in the distance r (= r ± ^/j dr), with an observed inten- 

 sity = e (= e ± Vî de) of a certain character is 



(1) Doir^drf{E)'^de. 



Integrating this expression between r = 0 and r = oo we obtain the whole 

 number of stars {=a{e)de suppose) in the cone having an observed intensity e 



a{e) = CO Z)(r)r- 1^ f{E) 



I. 



Û 



which I call the fundamental equation of stellar statistics. 



Ex. 1.) = if = luminosity of the star 



e = h = observed brightness 



.•.ff= hr\ 



dE dH „ , 

 — = — r = y , hence 

 de dh 



(a) a{h) = Ü) ldrD{r) r* ^(/jr^) 



0 



which formula gives the number of stars with the brightness h in a telescope 

 having a field equal to w. 



This formula first given (without I){r)) by Gtldén (1872), has been exten- 

 sively used by Seeligek (from 1898). 



Ex. 2.) = P = absolute motion of a star projected on a plane at right 



angle to the line of sight (short crossmotion). 



e = p = observed proper motion. 



• : P — pr. 



dE^dP^ 



de~ dp^ ' 



and 



(b) a{p) = (üjdr D{r) '-p^ipr). 



