529 



of stars of the apparent magnitudes 3.0 to 10.0 are found. These 

 are most reliable. 



The densities in the difïerent shells were determined by comparing 

 each time in two successive shells the numbers of stars of a deter- 

 mined absolute magnitude. The relative density' of two successive 

 shells was found as the average of three determinations, based 

 respectively on the stars of the apparent magnitudes from 3.0 to 

 8.0, 3.0 to 9.0 and 3.0 to 10.0. The density in the fifth shell, viz. 

 the one for which the mean parallax is 0".0296, was supposed to 

 be one, after which the density at every distance from the sun, 

 expressed in this unity, is known. 



The average density for the whole sky varies about in the same 

 way with the distance to the sun as Kapteyn found. In zone I the 

 density at great distances is considerably more than the average, 

 in zones III, IV and V on the other hand it is much less. 



By means of the densities found we now calculated from table 4 

 and the similar ones for the galactic zones the number of stars per 

 unit of volume, after the density had been reduced to the one for 

 rr == 0".029H. In order to do so each number in the last-mentioned 

 tables was dimijiished by the logarithm of the density of the shell 

 pertaining to it. In this way for instance table 5 has been calculated 

 from table 4. 



F'rom the tables that were found last of all, the luminosity curves 

 for the whole sky and the 5 zones may be deduced at once. The 

 numbers, standing between the fat lines in each column, correspond 

 pretty well. 



In table 5 we took the averages of these numbers (not of the 

 logarithms), and noted down the logarithm of these averages in the 

 last line of each table. In taking the mean equal weight was ascribed 

 to all numbers, except to those of the first four shells. These 

 numbers are not very reliable, because they are small, but 

 especially because we were obliged to exclude tVom our investigation 

 stars with a proper motion > 50" per century. Consequently the 

 luminosity curves that have been found are only of value up to 

 M = 5.0. 



It is of interest to point out that our result for the whole sky 

 corresponds beautifully to Kapteyn's found in /^z^^/. ("rywim^/^^n N°. 11. 



Furthermore it is remarkable that the curves found for the various 

 zones ditfer only a little. We already observed that from this we 

 may conclude with a certain probability that the luminosity curve 

 does not change with the distance to the sun either. 



In table 5 and the corresponding tables for the 5 galactic zones 



