873 



SCHWARZSCHILD SlippOSeS: 



D(r) = 10^0-"! P-«5P' {q = — 5.0 log. 7') 



tf{ï) =1 Oh-h M-b,.W (^M = — 2.5 log. i) 



i|'( V)= lO^o-'i G-C, G« ((j _ _ 5 o log^ V) 



In contradistinction to the assumptions of Seeliger about the form 

 of the densit}' law and the luminosity law these suppositions are 

 not without any foundation. The form for </• (/) has been found 

 empirically by Kapteyn and the form of the density function has 

 been derived in Astron. Nachr. N". 4422 by means of the numbers 

 of stars of determined magnitude that have been coifnted. The form 

 of the function \p ( I^) "ist zunachst rein formal der Bequemlichkeit 

 der Rechnung wegen eingeführt". Besides the formula (3) mentioned 

 above ScHWARZscHiLD deduced from the observations: 

 log.:;i,n = — 1.108 — 0. 125 m 

 log. .^„,^„= — 0.766 — R^m — 0.1243.<7 

 in which </ has been assumed to be =: — 5.0 log. ^i. 



From these data the three principal laws and various other 

 quantities, including the coefficient R^, were derived. 



According to Schwarzschit,d the distribution of luminosities found 

 by him may also be formulated in this manner, that the absolute 

 magnitudes are spread around the average value 11'". 5 (in Schwarz- 

 scHn,D's notation ^) with a mean error of 3'". 8 according to the law 

 of errors. This is incorrect. From the data used by him it is easily 

 found with the aid of the table of coefficients in Astron. jS^achr. 

 N°. 4557, that the mean M of his luminosity curve is 25'". 1, theie- 

 fore in Kapteyn's notation 30'". 1. 



The frequency curve found by Schwarzschild differs considerably 

 from the distribution of luminosities determined by Kapteyn in Fiibl. 

 Groningen N°. 11. This difference is puzzling as both investigators 

 made use of the same data. 



We have failed in finding a conclusive explanation for this bad 

 agreement. One might think, that a correlation between the absolute 

 magnitude and the velocity of the stars exerted its infiuence. Of late 

 several investigators have indeed drawn the attention to some indica- 

 tions of a relation between luminosity and velocity. "). But yet one 



'j Schwarzschild makes use of the distance corresponding with tt = 1" as a 

 unity of distance. Then M = m-\-h log. tt, so that the relation üfKAPTEVN = 

 = ■M'scHWARzscHiLD + 5 holds good. 



") Vide e.g. Eddington, Observatory Vol. 38, p. 392, seq. Perhaps this question 

 may also be solved by the investigation now in course of preparation at the 

 Astronomical Laboratory at Groningen and on which we already drew the attention 

 in our former communication. 



