1908] on Recent Researches in the Structure of the Universe. 309 



Staks of Measured 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 individual 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. 



?jrd Set. — Distance boxes. These percentages are all we want for 

 our last distribution, the distribution 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 d posteriori that even rather considerable uncertainties in 

 these percentages have but an unimportant influence on the results. 

 We are thus at last enabled to distribute our star-cards according to 

 the true distances. I made the distribution over the spherical shells 

 shown in Fisf. 2. 



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

 is removed from one shell to the next further one, the observer at the 

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

 it will grow very nearly 2^ times fainter. 



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

 results. The shells become too narrow towards the centre and the 

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

 figures. For this reason I constructed Fig. 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 oo years, and so on. In these 

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

 magnitude in each of the rows. The stars of the boxes of Fig. o are 

 thus, of course, all contained in the vertical row of boxes, correspond- 

 ing to apparent magnitude 5 in Fig. 4. 



Distribution according to Distance Illustrated 

 BY Example. 



In order to illustrate by an example how the stars of the boxes in 

 our Fig. 3 are distributed over our different shells, that is over our 

 distance boxes of Fig. 4, take the 7th box. It contains 77 stars at a 

 mean distance of 220 light-years. Our countings on the sample 



