( 627 ) 



which fairly represeni the numbers found in the conimunication 

 just mentioned. 



Let 



Apparent brlghtne'^s of a star of mag m 



rf =: 2 512 . . . = '- '- '^—- {log. 6= 0.4); 



„ „ m -f- 1 



H M ?? 



(_> z=i distance from the solar system (o = 1 for parallax r= 0"1) ; 



A'^,, = number of stars in the whole of the sky between the apparent 



1 1 

 magnitudes m — — and in -I . 



A ((>) =z star-density = number of stars per unit of volume at distance 

 Q (unit of volume = cube, each side of which = unit of distance). 

 Iii„ = apparent brightness of a star of the magnitude m (A5.5 = J). 

 Z> =: liuminosity = total quantity of light emitted (L = 1 for Sun). 

 ^/) (Z>) (/L = probability that the luminosity of a star/ chosen at 

 random, is contained between L and L -\- dL. 



(p {£) dz = probability that the luminosity is contained 



1 



between the limits L -\ mag. 



- 2 ^ 



Now, if we assume that (f {L) is not dependent on q, we shall have 



00 K [/(f 



N„, = 4^ I q' L{q) dQJ (f {hQ') dQ = i:r C q' A (9) ^ {h„,Q') dg (1) 



l/cf 



The expressions derived in Astron. Joiirn. W. 566 are : 



«^ . mod. 



yp{L) = g_^.2 [/„^Z-rp >2\ 



[/jtL ^ 



-^=e~^F-^i3ge~yf^ (3) 



in which 



T=z 1.400 

 a' — 0.385 



• (4) 



A (0) = 111.0 (5) 



i?= 0.0220 \ 



7 = 0.0052 ! ^^^ 



In a subsequent part of the same paper, the members of the stars, 

 as given by Pickering led to a new value of A (0) viz. 



A(0) = 136.9 (7) 



the difference of this value and the value (5) is wholly explained by 

 the constant difference of the photometric scale of Potsdam, which was 



43* 



