July 9, lyoSj 



NA TURE 



235 



¥ll6/'0 

 JflSi 



M]22tf 



77 



would be impossible to say in how far the results 

 definitively to be obtained would be influenced. Happily 

 there is an escape. 



For our last classification, the classification in the 

 distance-boxes, it is of no particular advantage that every 

 individual star gets in its proper distance-box. It will be 

 sufficient to know how many stars will finally be found 

 in each distance-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 investigate o sample of our stars. 



Stars 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 dis- 

 tributed 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 w-hich 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. 



2rd Set. — Distance ho.xes. These per- 

 centages are all we want for our last dis- 

 tribution, the distribution over the dis- 

 tances. 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 un- 

 important 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 Fig. 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 23 times 

 fainter. 



The figure is not well fitted for bringing 

 out the details of our results. The sfiells 

 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 hori- 

 zontal 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 draw- 

 ing. We thus see that the central sphere extends to a 

 distance of twenty-one light-years, that the second spherical 

 shell extends from twenty-one to thirty-three years, and 

 so on. In these rows a last set of bo.xes is placed. There 

 is a box for each apparent magnitude in each of the rows. 

 The stars of the boxes of Fig. 3 are thus, of course, all 

 contained in the vertical row of boxes, corresponding to 

 apparent magnitude five in Fig. 4. 



Distribution according to Distance Illustrated by E.xaniple. 



In order to illustrate by an example how the stars of 

 the boxes in cur Fig. 3 are distributed over our different 

 shells, that is, over our distance boxes of Fig. 4, take the 

 seventh box. It contains seventy-seven stars at a mean 

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

 showed that about one-fifth of the stars have true distances 

 which are between 37 per cent, and 59 per cent, of their 

 mean distance (derived from their apparent magnitude and 



NO. 2010, VOL. t8] 



proper motion). Therefore about one-fifth of our seventy- 

 seven stars must have true distances between 37 per cent, 

 and 59 per cent, of 220 light-years, that is, between 

 eightv-two and 130 light-years — or, finally, fifteen stars of 

 our box must find their place in the fifth shell of Fig. 4, 

 that is, in the box corresponding to the fifth apparent 

 magnitude in that shell. In precisely the same way I 

 find that twenty-one of them must be plai^d in the sixth 

 shell, eighteen in the seventh, ten in the eighth, and so on. 



If, after that, we repeat the process for all the remain- 

 ing boxes of Fig. 3, we get, for the fifth apparent magni- 

 tude, the numbers inscribed on the lower side of the boxes 

 corresponding to that magnitude in Fig. 4. 



Further than for the eleventh shell no numbers have been 

 entered. They become too uncertain. As, however, we 

 know the total number of stars of each apparent magni- 

 tude, we know the aggregate number which remains to be 

 distributed over the whole of the further shells. 



What has here been explained for the stars of the fifth 

 magnitude has been also done for the other magnitudes 

 between the second and the eightb The whole of the 

 results are shown in our Fig. 4. 





18S 



M]'286 



95 



M)248 



86 



MJ138 



m m 



■Ji 



m 159 



13-2 



WHO I 



79 



M)67 



4S 









Trf) 5S 



M)30 





f ^M)24 



M)22 





^M) 16 





i^WJIl 



J- L q 3 . 



Stars of Equal Luminosity brought together. 



The main result of the investigation is embodied in 

 these numbers — and first, in every box stars have now been 

 brought together of equal absolute' magnitude — that is, 

 of equal luminosity. For as the stars in each box are at 

 the same distance, and as, at the same time, they are 

 of equal apparent brightness, they must, of necessity, be 

 of equal total light-power, that is, according to our defini- 

 tion, of equal luminosity or absolute magnitude. For the 

 absolute magnitude of a star I have taken the magnitude 

 the star would show if placed at a distance of 326 light- 

 years. The choice of just this number is simply a matter 

 of convenience, and need not be explained here. 



As a consequence, the stars at a distance of 326 years, 

 which to us appear as stars of the fifth magnitude, will 

 have also the absolute magnitude five. Those of the same 

 apparent magnitude, but at a distance of 517 light-years — 

 that is, just one shell further — must have the absolute 

 magnitude four in order to show us the same brightness, 

 notwithstanding the greater distance. Now our eighth 

 shell lies just between these limits of distance. In the 

 middle of this shell, therefore, the stars of apparent magni- 



