12 



C. V. L. Charlier 



[JL > r'ooo 



for 



16 stars 



(X > 0.750 



» 



26 » 



[1 > 0 BOO 



» 



60 * 



[JL > 0.250 



» 



189 » 



[JL > 0.000 



» 



2616 » 



so that 



J(0:'500) : ^(i:'ooo) = 60 : 16 = 3.7B 



^(0."2ô0) : ^(0:'500) = 3.15 



^(0.125) : ^(0.2.öO) = (2.7) 



so that A(p) : A{2p) = 4 (and smaller) instead of 8 according to (9). For the mo- 

 ment we can draw no conclusion from this discordance. 



4. It is sometimes useful to have an expression for the number of stars of 

 a certain photometric magnitude m (instead of of a certain brightness h). Using 

 our fundamental formula I for 



E = M, e = m and observing that *) 

 w=-2.5log/i; M=~2.f,\ogH; H=hr^ 



which give 



log H = log h -\- 2 log r 



or 



(10) Ji"=m — 51ogr 



(the constant being so chosen that M=m for r= 1). 

 we obtain from I 



00 



(11) a(m) = iùj dr D[r) r'^ ^^(jm — 5 log r) , 



0 



where cpo(M) dll is the relative frequency of stars of the magnitude 31. Putting 



r — ^ •.• p = nat. logr 



the formula (11) may be wiitten 



(11*) a{m) = dp B(é')e^ '£j(m — 2 nui p). 



A more general assumption regarding D (than Z) = const ) is that discussed 

 by Seeliger (»Betrachtungen» 1898, 1909) 



(12) I) = ^r~\ 



Putting this value in the formula (a) of § 1 we get 



*) All lofrarithms used in this paper are referred to the base 10, if there is not expressly 

 stated that natural logarithms are concerned. 



