STRUCTURE OP THE UNIVERSE — KAPTEYN. 317 



STAR DENSITY FOR DISTANCES EXCEEDING T^VO THOUSAND LIGHT YEARS. 



Is there no possibility of getting beyond this distance? 



I think there is; but of course you Avill not be astonished to find 

 that the certainty of our conckision diminishes as we get deeper and 

 deeper in the aM'sses of space. 



One of the reasons why the method thus far applied breaks down 

 beyond the eleventh shell is that our data about proper motion are 

 not refined enough to determine this motion with sufficient accuracy 

 as soon as it is below 1" in a century. Even the somewhat greater 

 motions are rather uncertain. The proper motions thus can not help 

 us much beyond a certain distance. But we have still one valuable 

 element for the solution of our problem. This element is the total 

 number of stars separately for the apparent magnitudes. Thanks 

 mainly to the photometrical researches at the Harvard Observatory, 

 it has become possible to determine with considerable accuracy the 

 total number of stars of the first, second, etc., to the eleventh magni- 

 tude; with a fair degree of accuracy even those for the magnitudes 

 down to the fourteenth (inclusive). 



The density in the shells beyond the eleventh, not only for the stars 

 down to the eighth apparent magnitude, but, according to what has 

 been said a moment ago, also for the apparent magnitudes of 9, 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 figure 1 produces just 

 these totals for the corresponding apparent magnitudes. 



It can be proved that after the eleventh 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 w^e reach the limit of the stellar system. We can not 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 farther, than we can trust the truth of the 

 hypotheses which still underlie our reasoning. 



For future consideration thei'e thus remains the question, In how 

 far can we test the validity of these hypotheses? 



