STELLAR ASTRONOMY 407 



osity, that is, to the luminosity of the sun, there is no further 

 difficulty in transforming other absolute magnitudes into lumin- 

 osities. 



The result arrived at for the mixture-law by carefully working 

 out these ideas is roughly summarized in the following table: 



Within the sphere whose radius corresponds to the mean parallax of 

 the stars of the ninth magnitude there will be : 



1 star 100 000 to 10 000 times more luminous than sun 



46 stars 10 000 to 1 000 times more luminous than sun 



1 300 stars 1 000 to 100 times more luminous than sun 



22 000 stars 100 to 10 times more luminous than sun 



140 000 stars 10 to 1 times more luminous than sun 



430 000 stars 1 to 0.1 times more luminous than sun 



650 000 stars 0.1 to 0.01 times more luminous than sun 



The increase in these numbers, which for the very luminous stars 

 is extremely rapid, becomes slower and slower for the fainter stars. 

 It even seems as if we have to expect no further increase in the 

 number of stars having less than a hundredth of the sun's light. The 

 uncertainty of the extreme numbers, however, does not allow us to 

 assert anything very positively. 



Meanwhile we have introduced a new hypothesis, viz., that the 

 mixture-law is the same at different distances from the sun. 



By the overlap of the absolute magnitudes in the consecutive 

 shells, we have, to a certain extent, the means of checking the correct- 

 ness of this hypothesis. Thus, for instance (see Fig. 2), we find for 

 the proportion of the numbers of stars of absolute magnitude 5.4 

 and 4.4: 



by shell V, 



by shell VI, ^=3.53 

 255 



The numbers are slightly different; not more so, however, than 

 can be explained by the uncertainties of our data. On the other hand 

 we plainly see that by accumulation of such uncertainties the proof 

 of the identity of the mixture in largely distant shells must become 

 extremely weak. 



As a matter of fact, the conditions are not quite so bad as they 

 seem to be by Fig. 2, because we dispose of data for magnitudes other 

 than the fourth, fifth, and sixth there represented. In consequence 

 of this, the overlap of the absolute magnitudes in two consecutive 

 shells is much more considerable, and we have even some overlap 

 of non-adjacent shells. Still the proof of the identity of the mixture 



