408 ASTROMETRY 



at different distances from the sun will remain defective so long as 

 we have not at our disposal a sufficient number of proper motions of 

 stars of very different magnitude. 



If we had such data in sufficient number for stars from the third 

 down to, say, the fourteenth magnitude, then there would be an 

 overlap of not less than six magnitudes even for the first and the 

 seventh shell. We should then be fairly able to dispense with the 

 above hypothesis. 



As mentioned before, we need by no means despair of obtaining such 

 data in a near future. 



The distribution of the different degrees of luminosity in the 

 universe is not the only thing that can be derived from such data as 

 those shown in Fig. 2. 



For we can evidently also determine at once the number of stars 

 pro unit of volume, in other words the density, for any absolute 

 magnitude. For this purpose we have only to divide the number 

 of stars in any one shell by the volume of that shell. For the various 

 shells this volume has been inserted in the figure in a separate table. 

 I have given an example of this determination in Fig. 2. It is as 

 follows : 



Absol. mag. 4.4 Absol. mag. 3.4. 



055 i 55 

 Shell VI Density = -=20.4 =5.2 



771 211 



Shell VII Density = -=15.5 -^ = 4.25 



4y.7 4y.7 



Whence : 



Density VII 15.5 



Absolute magnitude 4.4 ... - = - -=0.76 



Density VI 20.4 



Density VII 4.25 



Absolute magnitude 3.4 ... -= - = 0.82 



Density VI 5.2 



Suppose the star-densities thus determined for all the absolute 

 magnitudes entering into our computations. If, as we assumed be- 

 fore, the mixture of the stars of different absolute magnitude is the 

 game throughout the system, then we must find the change in density 

 from shell to shell the same for every absolute magnitude. In our 

 example we find for the proportions of the densities in shell VII and 

 VI, 0.76 for the stars of absolute magnitude 4.4; 0.82 for those of 

 absolute magnitude 3.4. 



They are somewhat different, but not more so than can be ex- 

 plained by the defectiveness of our data. I found fairly the same 

 consistency for the whole of the materials. 



There thus provisionally is no reason for abandoning the hypothesis 



