STEUCTUEE OF THE UNIVERSE KAPTEYN. 311 



box. If this result is obtained, we shall presently see how easy it 

 becomes to study the problem put at the beginning of this lecture. 

 Our aim will be evidently reached if we can find out how many per 

 cent of the stars in any one box have such and such a distance. Now, 

 in order to determine these percentages, it will be sufficient to investi- 

 gate a sample of our stars. 



STARS OF 3IEASURED DISTANCE TAKEN AS A SAMPLE. 



Happily there is the possibility of taking a sample that will help 

 us out of the difficulty, for, as we know, there are in the sky a hundred 

 stars of which astronomers have succeeded in determining the indi- 

 vidual distance with some accuracy. We take these as our sample. 

 They are distributed over a great many of our boxes. 



We take them all out, having a care to note for all of them the 

 mean distance of the stars in the box to which they belong. For all 

 the hundred stars we now compare their mean distances to their true 

 distances, and thus find out how many per cent of them have true 

 distances between two and three, four and five tenths, and so on, of 

 the mean distance. 



Third set: Distance hoxes. 



These percentages are all we want for our last distribution, the dis- 

 tribution over the distances. It is true that our sample is a somewhat 

 undesirably small fraction of the whole; it shows, besides,. some other 

 weak points; but it appears, happily, a posteriori, that even rather 

 considerable uncertainties in these percentages have but an unimpor- 

 tant influence on the results. We are thus at last enabled to distribute 

 our star-cards according to the true distances. I made the distri- 

 bution over the spherical shells shown in figure 3. 



The dimensions of these shells have been so chosen that if a star is 

 removed from one shell to the next farther one, the observer at the 

 center will see the star grow fainter by just one magnitude; that is, 

 it will grow very nearly two and one-half times fainter. 



The figure is not well fitted for bringing out the details of our 

 results. The shells become too narrow toward the center and the 

 more central ones do not allow of the insertion of sufficiently clear 

 figures. For this reason I constructed figure 4. The numbers valid 

 for the several spherical shells have here been entered in equally 

 broad horizontal rows. The drawing does not, therefore, show the 

 real dimensions, but these as expressed in light years, which may be 

 read off on the right-hand side of the drawing. We thus see that the 

 central sphere extends to a distance of 21 light years ; that the second 

 spherical shell extends from 21 to 33 years, and so on. In these rows 

 a last set of boxes is placed. There is a box for each apparent mag- 

 88292— SM 1908 21 



