8 C. V. L. Charlier 



Ex. 3.) = A == projected distance between the components of a double star 

 e = S = observed d:o 

 •.• A = rS, and 



(c) a(§) = (1) jdr D{r) tp^IrS). 



Let F(x) be the frequency function of x (so that dxF{x) gives the probabi- 

 lity of a value of a: between the limits x ±''-/2 dx) then the quotient 



+ CO 00 



M{x) =Jx Fix) dx : / F{x) dx 



— 00 — 00 



gives the mean value of x (== M{x)). 



From our formula (1) giving the number of stars in the distance r ± Va dr, 

 we may thus derive the mean distance of a star having the character considered 

 in the intensity e ± ^/a de and obtain 



00 00 



II. Me{r) = j dr D(r) f{E) : | dr D(r) ~ f(E). 



0 0 



which formula may be immediatelj^ applied to our three examples above. 



2. Simplest assumption as to H and D. 

 Put 



D = const. ; ff'= const. 



which are the assumptions generally made in popular discussions concerning the 

 constitution of the Milky Way. 



From (2) H - hr^ = const. 



we now conclude that h is only dependent on r. The number of stars in an element 

 of volume is 



D(o r'^dr 



and from (2) 



n r 



hence the number of stars {a{h)dh) with the brightness h ± dh is 



D FC'^ 



(3) aih) dh — —~ (Ü —r- dh 



\ I 2 



Integrating we obtain for the number of stars A{h) with a brightness > h 

 the expression 



h 



(3*) A[h)={a{h)dh^-^ 



mi-' 



