310 Professor Dr. J. G. Kapteyn [May 22, 



showed that about one-fifth of the stars have true distances which are 

 between 37 and 59 per cent, of their mean distance (derived from their 

 apparent magnitude and proper motion). Therefore about one-fifth 

 of our 77 stars must have true distances between 37 and 51) per cent, 

 of 220 hght-years, that is between 82 and 130 light-years — or finally, 

 15 stars of our box must find their place in the 5th shell of Fig. 4. 

 that is in the box corresponding to the 5th apparent magnitude in 

 that shell. In precisely the same way I find that 21 of them must be 

 placed in the 6th shell ; 18 in the 7th ; 10 in the 8th, and so on. 



If, after that, we repeat the process for all the remaining boxes of 

 Fig. 3, we get, for the 5th apparent magnitude, the numbers inscribed 

 on the lower side of the boxes corresponding to that magnitude in 

 Fig. 4. 



Further than for the 11th shell no numbers have been entered. 

 They become too uncertain. As, however, we know the total number 

 of stars of each apparent magnitude, we know the aggregate number 

 which remains to be distributed over the whole of the further shells. 



What has here been explained for the stars of the 5th magnitude, 

 has been also done for the other magnitudes between the 2nd and 

 the 8th. The whole of the results are shown in our Fig. 4. 



Stars of Equal Luminosity brought together. 



The main result of the investigation is embodied in these 

 numbers — and first, in every box stars have now been brought 

 together of equal absolute magnitude — that is, of equal luminosity. 

 For as the stars in each box are at the same distance, and as, at the 

 same time, they are of equal apparent brightness, they must, of 

 necessity, be of equal total light-power, that is according to our 

 definition, of equal luminosity, or absolute magnitude. For the 

 absolute magnitude of a star, I have taken the magnitude the star 

 would show if placed at a distance of 326 light-years. The choice of 

 just this number is simply a matter of convenience, and need not be 

 explained here. 



As a consequence, the stars at a distance of 326 years, which to 

 us appear as stars of the 5th magnitude, will have also the absolute 

 magnitude 5. Those of the same apparent magnitude, but at a dis- 

 tance of 517 light-years — that is, just one shell further — must have 

 the absolute magnitude 4 in order to show us the same brightness, 

 notwithstanding the greater distance. Now our 8th shell lies just 

 between these limits of distance. In the middle of this shell, there- 

 fore, the stars of apparent magnitude 5 must have absolute magnitude 

 4*5. In the box, therefore, belonging to the 5th apparent magni- 

 tude, 8th shell, all the stars are of absolute magnitude 4*5. In 

 the 9th shell a star must already have the absolute magnitude 3 * 5 in 

 order to shine as a 5th apparent magnitude at this greater distance, 

 and so on. In this way the absolute magnitudes were found which 

 in our figure have been inscribed on the lids of the boxes. 



