. Annual report of the Board of Regents of the Smithsonian Institution . thereal dimensions, but these as expressed in light years, which may beread off on the right-hand side of the drawing. We thus see that thecentral sphere extends to a distance of 21 light years; that the secondspherical shell extends from 21 to 33 years, and so on. In these rowsa last set of boxes is placed. There is a box for each apparent mag-88292—SM 1908 21 312 ANNUAL EEPORT SMITHSONIAN INSTITUTION, 1908. nitiicle in each of the rows. The stars of the boxes of figure 2 arethus, of course, all contained in the vertical
. Annual report of the Board of Regents of the Smithsonian Institution . thereal dimensions, but these as expressed in light years, which may beread off on the right-hand side of the drawing. We thus see that thecentral sphere extends to a distance of 21 light years; that the secondspherical shell extends from 21 to 33 years, and so on. In these rowsa last set of boxes is placed. There is a box for each apparent mag-88292—SM 1908 21 312 ANNUAL EEPORT SMITHSONIAN INSTITUTION, 1908. nitiicle in each of the rows. The stars of the boxes of figure 2 arethus, of course, all contained in the vertical row of boxes, correspond-ing to apparent magnitude 5 in figure 4. DISTEIBUTION ACCORDING TO DISTANCE ILLUSTRATED BY EXAMPLE. In order to illustrate by an example how the stars of the boxes inour figure 3 are distributed over our different shells, that is over ourdistance boxes of figure 2, take the seventh box. It contains 77 starsat a mean distance of 220 light years. Our countings on the sampleshowed that about one-fifth of the stars have true distances which are. Fig. 3.—Distance Boxos. between 37 and 59 per cent of their mean distance (derived from theirapparent magnitude and proper motion). Therefore about one-fifthof our 77 stars must have true distances between 37 and 59 per centof 220 light years; that is, between 82 and 130 light years; or, finally,15 stars of our box must find their place in the fifth shell of figure 1;that is, in the box corresponding to the fifth apparent magnitude inthat shell. In precisely the same way I find that 21 of them must beplaced in the sixth shell, 18 in the seventh, 10 in the eighth, and so , after that, we repeat the process for all the remaining boxes offigure 2, we get, for the fifth apparent magnitude, the numbers in-scribed on the lower side of the boxes corresponding to that magni-tude in figure 4. STKUCTUEB OF THE UNIVEKSE KAPTEYN. 313 Further than for the eleventh shell no numbers have been become too unce
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