526 



zones separately, this difficulty is diminished considerably. We can 

 then compare the luminosity curves, found for the various zones. 

 If now, it appears that these curves correspond, it is highly impro- 

 bable that the distribution of luminosities varies with the distance 

 to the sun. If the luminosity law depends on r and not on b, we 

 should find a fan-shaped conjposition of the sidereal system, which is 

 not inconceivable, but highly improbable all the same. 



If we may assume that the distribution of the luminosity is 

 independent of the distance to the sun, Kapteyn's method enables 

 us to determine the absorption, or at least to examine if it has any 

 influence on the distribution of stars in space which we found. 



The thiid hypothesis was, indeed, when Kaptkyn made it for the 

 first time a rather arbitrary assumption, and it must be conceded 

 that other results would have been found, if another hypothesis had 

 been made. The hypothesis was made, because at that time there 

 were still too few measured parallaxes, to enable us to deduce the 

 form of the frequency curve log. ""/an directly from the data. Perhaps 

 this will be possible when applying the method once more. It is, 

 however, of little importance to discuss in the present stage the 

 question whether this hypothesis could be justified or not, as 

 ScHWARZscHiLD has proved that the above-mentioned relation exists 

 for a special form of the density law and the luminosity law, which 

 form obtained a great amount of probability owing also to his 

 investigations. 



We have deduced the luminosity law from the data of observation 

 mentioned in \ 2. Space was divided into a number of shells the radii 

 of which had been selected for convenience' sake in such a way 

 that lay. r increases with 0.2. Afterwards the mean parallax was 

 determined for eacli of the numbers A'„yy thai had been found by 

 means of the formula for .t,„ „. Then with the aid of the value 

 found for the probable deviation of the error curve log. ~'l-a,,, it was 

 calculated which part of the Jiumbers found occur in every shell. 



The numbers iV,„,„ which we found for the whole sky are given 

 in table 1. and the corresponding parallaxes .t„, „ in table 2. In 

 table 3 we have mentioned, how these numbers of stars are divided 

 over the various shells. Tables in accordance with this we have 

 calculated for the five zones. 



Now we have derived from these tables others, indicating the 

 numbers of stars of every absolute magnitude per unit of volume 

 in the different shells. For the whole sky we communicated our 

 results in table 4. Between the fat-faced lines the numbers 



