Studies in Stellar Statistics 21 

 corresponding to a parallax (n) = 0."i, to liave the magnitude 5"'.B. we hence have 



(46) ilf =— 2.5log* // + 5.B 



or 



(46*) log jg"= 2.2 — 0.4 Jf. 



Introducing the natural logarithm instead of ^"log we thus have 



(46**) H = 158.5 e~^*^ where = + 0.460517 = 0.2 X ^ 



xMod. ■ 

 and also 



(46***) Ä= 158.6 . 



Introducing these values in (45) we have 



(45*) ^,{M) = 6\ "'^"-^ - 



or 



The exponent has a maximum for 



0 = 26 + 0.32 a2 (2.0 - M). 



= 0.92108 4- 0.1232 (2.0 — M) 



or 



M= 9"\476 



which hence is the mean magnitude of the stars at the distance z = 0'' 1. 



The frequency function a{m) for the apparent magnitudes has according to 

 Kapteyn and Schwarzschild (1. c.) the expression 



log fl(m) = 0.596 + 0.5612 m — 0.0055 



or 



(47) ■ nat. log a{m) = 1.372 + 1.2922 in - 0.01266 



The value of m corresponding to the maximum value of a{m) is given by 



0 = 0.5612 — 0.011 m V m = 51'". o 

 and the expression of a{m) in the form (41) is 



a[m) = ce ' (Wq = 51.0) 



or 



y-^^l a{m)=ee , = 6™.285 (Ä:^ = 39.50) 



a rather startling result! If the results of Kapteyn be true, the number of the 

 stars should increase to the ôl:th magnitude 1 The total number of the stars as 



