874 



may certainly not assume, that such a relation would be of so far- 

 reaching importance and would liave so great an influence on the 

 result in applying Schwarzschii.d's method. Moreover, treating the 

 data of observation available to us according to this method we found 

 the same luminosity curve we had found according to Kapteyn's method. 



As tiiere is no agreement, it seems to us, that Kapteyn's result 

 should be trusted most. His method is, indeed, to be preferred to 

 Schwarzschild's and this chiefly for the following reasons: 



1''*. Kapteyn does not suppose definite forms for tiie functions 

 that are to be determined ; 



2"^'. with Kapteyn's method it is possible to examine, if the fre- 

 quency function of absolute magnitudes is the same at all distances 

 from the sun ; 



3"^ if Kapteyn's method is applied, it may be seen at once how 

 far the results are based on observations and where extrapolation 

 comes in ; 



4'^. Kapteyn's method need not be altered if it should appear 

 that velocity is a function of luminosity, while Schwarzschild's 

 formulae must undergo considei-able modifications, and 



5f''. Schwarzschild's results depend to a high degree on the values 

 of some quantities (e.g. the coëfficiënt «,) which can only with diffi- 

 culty be derived with sufïicient exactness from observations. 



Yet Schwarzschild's method and Kapteyn's if they are applied 

 prudently to the same data, should give the same results. The 

 difference found is probably to be attributed to the fifth objection 

 that we raised against Schwarzschild's method. 



The interpolation-formulae we deduced from our data for jV,„ and 

 the formulae used by us for .t„, and jr,,,^,^ are to be trusted more 

 than Schwarzschild's, because they are based on more complete 

 and more minute data. Therefore less danger is to be apprehended 

 from the objection mentioned for our determination. So it may 

 perhaps be explained that Schwarzschild found a divergent result, 

 as we attained the same results according to the different methods. 



2. The results of our investigation. 



We have applied vSchwarschild's method to the same data of 

 observation that we have also treated, as we have communicated 

 in a former article, according to Kapteyn's method. 



The data we want are : 



1. the numbers of stars of determined magnitude A',„, 



2 the mean parallaxes of stars of determined magnitude .t„,, 



3. two coefficients of the formula for :7t,„^^. 



