312 Professor Dr. J. G. Kcqiteyn [May 22, 



This table represents what, up to the present time, we know 

 about the mixture-law. 



The fainter the stars, the more numerous. 



The rate at which the numbers increase with the faintness is 

 particularly noticeable for the very bright stars. 



Passing to the fainter stars, this rate gradually diminishes, and it 

 looks as if we must expect no further increase in number for stars 

 whose luminosity falls below one-hundredth of that of the sun. 

 Meanwhile this is simply a surmise. For stars of this order of 

 faintness data begin to fail. Here, as in nearly every investigation 

 about the structure of the stellar system, the want of data for stars 

 below the 9th apparent magnitude makes itself very painfully felt. 



But let us come back to our Fig. 4. I will first remark that, 

 knowing the mixture-law, we can predict the number of stars that we 

 shall get in the empty boxes belonging to the 9th, 10th, etc., magni- 

 tude, as soon as continued astronomical observations will permit us 

 to include these stars in our discussion. For the mixture-law, as 

 derived just now, shows that in our universe the stars of absolute 

 magnitude 5 * 5 are 3 ' 5 times as numerous as the stars of absolute 

 magnitude 4-5. 



Now as in the 11th shell the number of stars of the absolute 

 magnitude 4-5 is 5400 (see. Fig. 4), there must be 3*5 times 5400, 

 that is, 18,900 stars of absolute magnitude 5*5 iii this shell. These 

 belong all in the box of the 9th apparent magnitude of this shell. 

 In the same way we obtain the numl^er of stars to be expected in 

 the boxes of the 10th, 11th, etc., apparent magnitude for all our shells 

 down to the 11th. There is exception only for the boxes belonging 

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

 14-5. 



It is evident, however, that the number of stars in these excej)- 

 tional 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. For 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 9th and 

 10th shells (see Fig. 4). In the 9th there are 49 stars of absolute 

 magnitude 2*5. Therefore, if in the 10th the stars were as thickly 

 crowded as in the 9th, there would occur in this shell four times 49, 

 that in 196 stars of this absolute magnitude 2 '5. 



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



140 

 dently must be that tbe star-density in the 10th shell is about ^ q^, that 



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