54 



C. V. L. Charlier 



of this function, for the present we must confine ourselves to the attempt to deter- 

 mine it indirectly from the fundamental equations. 



There remains the function Me{r). The possibihty of determining the mean 

 distances of the stars for different magnitudes directly from parallax-measurements 

 is, at the present state of observational astronomy, nearly excluded. With the 

 help of other characteristics — the proper motions and the distances between the 

 components of double stars — we may, however, arrive at an approximate solution. 

 Even then we must however have recourse to hypothetical, though plausible, con- 

 siderations. We are, indeed, in different ways, of which some have been considered 

 in these lectures, led to the conclusion that the mean distances of the stars of the 

 magnitude m ± Vs dm follow the law 



M(r) = k/^^"*' 



where and Xj designate certain parameters. This conclusion assumed, we have 

 to determine Xj from the proper motions and from the distances between the com- 

 ponents of double stars, whereas is to be found from the direct parallax- 

 measurements. 



Both of these determinations seem to be connected with very great uncertain- 

 ties. Using the proper motions in Boss Preliminary Catalogue of Stars I have 

 deduced a value of Xj amountiug to about -f 0 33. The parallaxes obtained by 

 Kapïetn — which are often used by the astronomers — lead, however, to a value 

 double as great as that. I have been unable, for the present, to reconcile these 

 results. 



As to I have still made no investigations for detennining it from the 

 actual parallax-measurements. Simply as a basis for discussion I have, in agreement 

 with usual assumptions, made the preliminary supposition that distance of the stars 

 of the 5th magnitude is equal to 10 siriometers, from which assumption the value of 

 can be obtained, when Xj is known. It must be observed that theoretical reasons 

 make it probable that the parameter has a different value in different parts of 

 the sky. 



Working out the problem of the constitution of the Milky Way according to 

 the scheme, shortly sketched above, it was found necessary to treat each part of 

 the heaven separately. Most of the pnrameters of our functions vary indeed greatly 

 from one part of the heaven to the other. Especially may it be noted that a 

 combination of observations from the whole sky must necessarily lead to an erro- 

 neous determination of the density function. We are nevertheless not prevented 

 from deducing results of fundamental importance even in this manner. 



Hence I have divided the sky into 48 parts of equal area, according to the 

 diagram in fig. 2. For each part (called »square») I have collected — and am 

 collecting — as full information as possible regarding the brightness and proper 

 motions of the stars and regarding the distances between the components of double 

 stars. The material regarding the doubles is in the hands of Mr John M. Pook of 



