880 



objections may be raised against the general validity of this theorem. 

 Sremger thought the premise of his theorem was affirmed by obser- 

 vations. This conclusion was premature and appears to be incorrect 

 by comparison with more exact data. Therefore the density law 

 derived by Seeliger cannot be accepted. And also his considerations 

 on a limit of the sidereal system found by him fail in sufficient 

 foundation. 



Seeijger determined the luminosity law by solving an integral 

 equation well-known in stellar astronomy by means of the density 

 found. No great significance can be attributed to the result derived 

 in this way. 



For the luminosity law, we indicated by "Seeliger I", use was 

 made of the density law D{r) = y r~'\ while in establishing the curve 

 "Seeliger II" he repi-esented the density by Z)(r) = y— '— « 7'-^i. 

 No sufficient motives have been stated for this last form. Moreover 

 n both determinations a definite form of the function 7(1') was pre- 

 supposed, which is undesirable and unnecessary. 



The curve in our figure with the indication "Seeliger III" has not 



been deduced by Seeliger; but it is a consistency of his theory. If 



we extend namely — as Prof. Kapteyn observed — Seeliger's 



theorem to stars fainter than the limiting magnitude '), then we find : 



/— 3 3—/ 



If, for m ^ 11, Am = Ch,n - , then ff{i) = Ai ^ independent of D{r). 



We now find according to Pxihl Groningen N°. 27 for 12.0 < m < 16.0 

 with some approximation : 



log. A,n = 1.797 + 0.340 (m— 12). 



This gives the luminosity curve (f{i) = Ai^-^^, which we indicated 

 in the figure by "Seeliger III". 



The graphical representation indicates how much the frequency 

 curves found by different in\ estigators differ mutnally. All the more 

 it is of importance to observe that our independent investigation 

 furnished a perfect affirmation of the luminosity law deduced by 

 Prof. Kapteyn in 1901 from the data still so scanty at that time. 



Kampen, September 1918. 



•) The value of n is according Seeuger about 11.5. 



