studies in Stellar Statistics 



13 



a{h) = ^(A fdr * ^{hr^), or 



0 



(12*) a{h) = = ^ (C = const.). 



h 2 



i. e. if D is proportional to a certain power ( — s) of r, then a{h) is propor- 

 tional to a power ( — x) of where 



Using (11*) it follows from (12) that 



(13) a{m) = C, ^™ = C, e'"' (v = (3-s) 0.46054) 



In the researches of Seeligee on the constitution of the Milky Way the forrn 

 (12) for D plays an important rôle. I will in the next of these memoirs discuss 

 more fully the methods of Seeligee,. Here I shall only draw some consequences 

 of the formula (12) without taking into consideration certain other assumptions used 

 in the researches of Seeligee. 



5. Even if the observations should show the formula (12*) for a[h) to be 

 valid for all values of h, one would not be allowed to conclude that the density 

 in the Milky Way follows the law (12). The equation 



00 



(14) a{h) = lûfâr D{r) r* tp{hr^) 



0 



connecting a(h) with D and f certainly now has the solution (12), but there are 

 other solutions equally possible. It will suffice to take out the following one 



which evidently gives 



a{h) = C-.K', 



whatever value may be given to the function D(r). 



The observations may show that the brightness follows the law (12*) more or 

 less approximately, but from that fact we are by no means entitled to declare that 

 the density must follow the law (12) not even approximately. Any law for the 

 density is possible, if the frequency function tp for the absolute brightness (lumino- 

 sity) of the stars is given by (15). Nor is it certain that these two solutions ((12) 

 and (15)) are the only possible ones. Probably there is an infinity of solutions 



Before leaving the hypothesis of Seeligee (12) in regard to the density in 

 the Milky Way let us examine its bearing upon the proper motions and on the 

 mean distances. 



