316 ANNUAL REPORT SMITHSONIAN INSTITUTION, 1908. 



same way we obtain the number of stars to be expected in the boxes 

 of the tenth, eleventh, etc., apparent magnitude for all our shells 

 down to the eleventh. There is exception only for the boxes belong- 

 ing to the lower shells, for which the absolute magnitude would 

 exceed 14.5. 



It is evident, however, that the number of stars in these exceptional 

 boxes must be small, and for what follows they are of little 

 importance. 



STAR DENSITY. 



In the second place, our boxes now also lead to the determination 

 of the star densities. IFor the volumes of the consecutive shells are 

 perfectly known; they are in the proportion of 1 : 3.98, For the 

 sake of convenience, let us say that the volume of each shell is exactly 

 four times that of the next preceding one. Now", to take an example 

 of the determination of the densities, consider the ninth and tenth 

 shells (fig. 4). In the ninth there are 49 stars of absolute magnitude 

 2.5. Therefore, if in the tenth the stars were as thickly crowded as 

 in the ninth, there would occur in this shell four times 49, that is, 196 

 stars of this absolute magnitude 2.5. 



In reality we find but 140 of these stars. The conclusion evidently 

 must be that the star density in the tenth shell is about lie 5 that 

 is about two-thirds of that in the ninth shell. A similar conclusion 

 is obtained by comparing the number of the stars of absolute magni- 

 tude 3.5 in the two shells. The values obtained from the magnitudes 

 0.5 and 1.5 may be neglected. Owing to the exceedingly small 

 number of stars, they must necessarily lead to untrustworthy results. 

 From all the rest I found that the density in the tenth shell must be 

 about G4 per cent of that in the ninth shell. The proportion be- 

 tween the densities in the other shells was determined in exactly the 

 same way, 



A slight defect in our results was then discovered. 



We should exceed the limits of the time allowed for this lecture 

 by entering into a consideration of this defect. It must be sufficient 

 to state that it was not difficult to remove it. After that it appeared 

 that the density in the first six of our shells is nearly the same. 

 The desnity in these shells, that is, in the neighborhood of our sun, 

 is such that about 2,000 stars of a luminosity exceeding one-hundredth 

 that of the sun must be contained in a cubic light century. After the 

 sixth shell the density diminishes gradually at such a rate that in the 

 eleventh shell the density has fallen to about 30 per cent of what it is 

 in the vicinity of the solar system. 



In what precedes we tried to give a solution of the problem put at 

 the beginning of this lecture — a solution, however, which embraces 

 only that part of the universe which is contained within a distance of 

 about 2,000 light years from our solar system. 



