314 Professor Dr. J. G. KaiiUyn [May 22, 



10, etc., to 14, has to be determined in such a way that the addition 

 of all the numbers in any one vertical column of Fig. 4 produces 

 just these totals for the corresponding apparent magnitudes. 



It can be proved that after the 11th shell the density must, on 

 the whole, continue to diminish. If we assume that this diminution 

 is gradual and proportional to the increase in distance, it becomes 

 very easy to determine the rate of this diminution, and consequently 

 the distance at which the density becomes zero, that is the distance 

 at which we reach the limit of the stellar system. We cannot enter 

 into fuller particulars here. It must be sufficient to say that in this 

 way we are led to conclude that the further diminution of density 

 must be slow, so slow that in the assumption made above, the limit 

 of the system is only reached at a distance of some 30,000 light- 

 years. 



Hypotheses Underlying the Results. 



In conclusion, a few words on the question : In how far are the 

 results now obtained to be considered as established ? 



The answer must be : They can be considered to be established 

 only in so far, and no further, than we can trust the truth of the 

 hypotheses which still underlie our reasoning. 



For future consideration there thus remains the question, in how 

 far can we test the validity of these hypotheses ? 



These hypotheses are the following : 



1. The mixture was assumed to be the same at greater and 

 smaller distances from the solar system. 



2. The same was done for different distances from the galaxy. 



3. The universe was assumed to be transparent, that is, it was 

 assumed that the absorption of light in space is zero. 



Can we get rid of these hypothetical elements ? 



I think we can, at least to a very great extent. 



As to the first. Our figure 4 already goes far in enabling us to 

 judge whether it is true or not. For evidently both our Hth and our 

 Oth sliell give the nature of the mixture, at least of the stars of 

 absolute magnitude 3*5 to 6*5. Therefore, as far as these stars are 

 concerned, we are able to see whether or not the mixture is the same 

 at the distance of 650 light-years as it is at the distance of 170 light- 

 years. Likewise the figure enables us to make the comparison in 

 other cases. As soon as we possess the necessary data for a longer 

 range of apparent magnitudes, say down to the 14th or 15th, we 

 shall be able to dispense to a very large extent with our first 

 hypothesis. 



As to the second, the possible variation of the mixture with the 

 distance from the Milky-way, it is largely only the question of 

 treating the stars in different galactic latitudes separately. As far 

 as I can see there are no particular difficulties in the way of such a 



