studies in Stellar Statistics 



49 



l)nt will iu stead carry out the remainiug computations with three different values 

 ol; X,, uan^ely Xj = ^ 3, X, = V2 and \ = Vs. 



26. According to our formulae given in the 14th § and using the units of 

 the 21th § we start from the following two equations 



N 



(76) 



(76*) 



a{m) = 



2J;2 



— X,6m 



Using the distances themselves — expressed in Siriometers — instead of the 



parallaxes, the last formula may be written 



(76**) M„,{r) = K, e 



As to the value of (or K^, its value must for the present be considered as 

 rather uncertainly known. When determining it from the observed parallax values, 

 it is always, as far as I know, presumed that its value is constant all over the 

 heaven, which is surely not the case. For want of better information I will use 

 the same value of (and in all squares and assume the mean value of the 

 distance for stars of the 5th magnitude be equal to 10 Siriometers (parallax 0!'o206), 

 which nearly coincides with the usually adopted value. Then (76**) may be written 



(77) 



so that 

 (77*) 



Xj h{m — 5) 



M{r) = 10 e 



log M{r) --= 1 — Xj + 0.2 



Tab. 11. M[r) in Siriometer. 



With this formula the following table is 

 computed which gives the mean distances of the 

 stars of different magnitudes (from — 1 to 20). 

 It is plainly shown how considerably the uncer- 

 tainty as to the value of Xj influences our know- 

 ledge of the true distances in the Milky Way. 



When a(m) and M[r) are supposed to be 

 given the values of the functions t^J^y) and 's^iy) 

 are to be found from the integral equations 



(78) 



+ 00 



a{m) = jdy A^iy) <Po(m + y) ; 



+ 00 



a{m)M4r) = fdy e '^A^Cy) <p,{m + y). 



+ 00 



I recapitulate here the solution of these equations 

 of the form 



Lunds Univ. Årsskrift. N. F. Afd. 2. Bd 8. 



m 



\ = 'A 



\ = 



X, = -Va 



— 1 



4.0 



2.5 



1.6 



0 



4.6 



3.2 



2.2 



1 



5.4 



4.0 



2.9 



2 



6.3 



5.0 



4.0 



3 



7.4 



6.3 



5.4 



4 



8.6 



7.9 



7.4 



5 



10.0 



10.0 



10.0 



6 



11.7 



12.6 



13.6 



7 



13.6 



15.8 



18.5 



8 



15.8 



20.0 



25.1 



9 



18.5 



25.1 



34.1 



10 



21.5 



31.6 



46.4 



11 



25.1 



39.8 



63.1 



12 



29.3 



50.1 



85.7 



13 



34.1 



63.1 



117 



14 



39.8 



79.4 



158 



15 



46.4 



100 



215 



16 



54.1 



126 



293 



17 



63.1 



158 



398 



18 



73.6 



200 



541 



19 



85.7 



251 



736 



20 



100 



316 



1000 



Suppose first Aq and tp^ to be 



