879 



that we found according lo tlie same method and marked bj the 

 figure I. Whereas the first is entirely different from Kapteyn's lumi- 

 nositj' law, the second agrees very well with it. 



In Astro?!. Nnchr. N". 4422 Hertzsprung assumed for the distri- 

 bution function of absolute magnitudes a Gaussian curve with a 

 mean value of 2'". 7 and an average deviation of rb 3"'.0. 



Halm ^) has assumed in his establishing of the luminosity law 

 that the density is constant. He also supposes a perceptible extinc- 

 tion of the light in space. With the aid of these hypotheses the 

 luminosity curve was deduced from the numbers of stars of deter- 

 mined magnitude found by Chapman and Melotte and the mean 

 parallaxes of Kapteyn and Comstock. It is remarkable that the 

 curve found in this manner agrees pretty well with Kaptkyn's. 



In Monthly Notices Vol. 72 Dyson has published an investigation 

 founded on the cross components of the stars of Carrington's Cir- 

 cumpolar Catalogue. Supposing that the density in the space taken 

 up by these stars is everywhere the same, he determined the lumi- 

 nosity law. The curve found in this manner has been drawn by us. 



Comstock '-) and Walkey ') have derived the frequency function of 

 absolute magnitudes from the luminosities of stars the parallax of 

 which has been measured. 



In his investigations on the structure of the universe Seeliger 

 has established the density law in the first place. This determination 

 rests on the following theorem found by him^): 



/— 3 



If, for /// <^ n, A,u = Chm ^ , the density D will be = yr~' what- 

 ever (/)(/) may be. 



Here A,n means the number of stars from the brightest star to those 

 of the magnitude in and A„, means the brightness of the stars of the 

 apparent magnitude m, while r represents the distance from the sun. 



We may formulate this theorem of Seeliger also in this manner: 



If the numbers of stars of determined magnitude form a geome- 

 trical progression, the density is proportional to a negative power of r. 



We have proved in oui' thesis for the doctorate ^), that several 



') Monthly Notices, Vol. 77. 



') Astron. Joiirn. N», 569. 



«) Astron. Nachr. N». 4754. 



+) The demonstration given by Seeliger is very intricate. We have published, 

 however, in our doctoral dissertation a very simple proof, which we owe to 

 Prof Kapteyn. 



5) On the Determination of the Principal Laws of Statistical Astronomy. Amster- 

 dam, KiRCHNER. 1918. 



57* 



