studies in Stellar Statistics 



23 



The mean distance of stars having the magnitude ni is according to (22) given 



bv the formula 



+ 00 



aim) M,n{r) = / chj Hy)'f^(m + y) g-"^ 



—00 



where A(f/) is given by (50^ and cp^ by (49). Applying the formula (52) we evidently 

 have to put 



s 



M. 



m 



o ' 



31,^31^- 



hence we get according to (21) and (51) (the values of the constant factors are 

 irrelevant) 



a(w) = Cj e 



(m— iWo)» 

 2*2 



as before, and according to (52) 

 (53) 



3Irrir) = a e 



jt2 



The term of the second dimension in m in the exponent vanishes. Comparing 

 this formula with (19), which was obtained from the hypothesis of Seeliger, we 

 find the mathematical expression for .l/m(r) in both cases to be similar, but the 

 coefficient of m in the exponent differs by the factor 

 (p_^2) : 7^2 



Using the values of h and K obtained above and 

 assuming the parallax of stars of the 4th magnitude to be 

 equal to 0''025i we get the values of tt shown in the ac- 

 companying table. 



The mean distance of the stars of the 51st magni- 

 tude — v/hich should represent the most numerous class 

 of stars according to (48) — should be 10 000 000 units 

 (a unit of distance corresponding to a parallax = 0!'i)! 



Let us next consider the density. We have according 

 to (50) (and the following lines) for the density, at the 

 distance r, the expression 



(2/-I'o)2 



where 



m 





1 



1 1 



0.075 



2 



0 052 



3 



0 036 



4 



(0.0^51) 



5 



0.0174 



6 



0.0121 



7 



0.0084 



8 



0.0058 



9 



O.OOlO 



10 



0.0Ü28 



14 



0.00065 



18 



0.00015 



51 



0.000 000 000 86 



0 — 5.60. 



(54) Z)(r) = Ce _ 



r„ = l,83; r = e~'^ 



The relation between r and y may also be written (we have h = 0.2/loge) 



log r = — 0.2?/, or 



54* 



y = — 5 log r. 



Putting here r=l, 10, 100, 1000 etc. and using the notation z ^ {y~l\) . o 

 we get the values of y from (54*) and those of D[r] from (54) with the help of 



