877 



Zone V. 

 log. D{r) = - 4.207 — 0.362 q — 0.0254 q* 

 log. <f{i) = — 0.636 + 0.045 J/— 0.0621 M^ 

 log.mV)— — lAS6 -0.784 G — 0.2158 (?' 

 il/,„ = — 6.4 -f 0.290 m 



The values we found for the various coefficients differ rather 

 considerably from Schwarzschii.d's results. This is especially apparent 

 in the formulae for rp(i) and i^^(V). 



It is especially of importance to compare the luminosity curve 

 with that of Schwarzschild and the frequency curves that wei-e 

 found for the various zones with each other. To facilitate this we 

 have given another form to the formulae. In order to compare them 

 we have also given Kapteyn's results in the same form. Here too 

 M has been expressed in the unit, used by Schvvarzschii.d. 



We represent the luminosity curve by the formula: 



A simple calculation shows us the relation between the new 

 parameters and those used above. 



The luminosity curves found above may be expressed now by 

 formulae of the form we found, with the following values of the 

 parameters : 



k h 



The whole sky . . 1.76 0.385 



Zone I . . 

 Zone II . . 

 Zone III . . 

 Zone IV . . 

 Zone V . . 

 Schwarzschild 



2.00 0.378 



1.61 0.382 



0.21 0.469 



0.51 0.456 



0.36 0.478 



25.07 0.184 



I type I 3.95 0.243 



Kapteyn ^) I ^^^p^ jj ^ 3() (^247 



The difference between the values found by us and those of 

 Schwarzschild is very great, much greater than we had expected. 

 There is more agreement with the results obtained by Kapteyn, 

 although here too at first sight the difference is pretty conside- 

 rable. No great significance can be attributed, however, to the \alues 

 found for k. These results are based on extrapolation as the parts of 

 the frequency curves that are based on observations do not extend as 

 far as the maximum. It appears from the figure that we added to 

 this communication, that Kapteyn's curve does not differ much from 



Astron. Journ. N". 566. 



57 

 Proceedings Royal Acad. Amsterdam. Vol. XXI. 



