RECENT ADVANCES IN SCIENCE 359 



space in regard to numbers and luminosities. The two laws 

 which govern the distribution are the density law — ^which gives 

 the number of stars per unit volume of space in different parts 

 of the system — and the luminosity law — which gives the pro- 

 portion of stars between different limits of absolute brightness. 

 It is generally tacitly assumed that the density law is the same 

 for all luminosities, and that the luminosity law is the same at 

 all distances. These assumptions considerably simplify the 

 mathematical discussion, and enable a satisfactory representa- 

 tion of the observational data to be given within the limits of 

 possible error. The data referred to are the counts of the 

 numbers of stars between given limits of magnitude, and the 

 mean parallaxes of stars of given magnitudes. 



The validity of these assumptions has been discussed in a 

 recent paper by J. Halm {M.N., R.A.S., 80, 162, 1919). He 

 combines the two laws into one function, which he calls the 

 distribution function F, defined so that the number of stars 

 in a cone of solid angle d(o, with its apex at the sun, situated 

 between the distances r i ^dr, and whose apparent magni- 

 tudes are between m dz i dm, is given by 



dN = F r^drdmdco 



where F is a function of r, m and the galactic latitude and 

 longitude. Since the absolute luminosity, M, is a function only 

 of m and r, F can be regarded also as a function of r, M and the 

 galactic latitude and longitude. 



A particular form of the expression F is F{ar,h{M -{- A)], 

 where a and A are functions of position, and /f is a constant. 

 Halm shows that this special functional form is in agreement 

 with observational data. The general assumptions referred 

 to above are equivalent to a still further specialisation, in which 

 F is assumed to have the form 



F=/(ar)t[/z(M + ^)] 



where / is a function of r, and the galactic latitude and longi- 

 tude, but is independent of M, whereas -v/^ is a function M only, 

 and is independent of the space co-ordinates. 



Halm then proceeds to show that this assumption does not 

 lead to the only forms of / and "^ which satisfy the statistical 

 data. He considers an alternative possibility, in which the 

 density is constant throughout space, so that / is a constant, 

 whilst the luminosity law is different at different distances ; 

 i.e. ^ is a function of the position. These suppositions can be 

 made to satisfy the counts of stars of different magnitudes 

 absolutely, and the mean parallaxes of stars of a given magni- 

 tude are the same as those obtained on the more usual assump- 

 tions to within the limits of observational error. 



