237 



M = . R 



If we take for M the mass of our galactic system, which can be 

 estimated ') at i • 10'" (sun = 1), then the last formukx gives R = 41, 

 or only about 1'/, times the distance of Neptune from the sun. This, 

 of course, is absurd. If we use the otiier formula we can take for 

 Qg tlie star-density in the immediate neiglibourhood of the sun, which 

 we estimate at 80 stars per unit of vohime of Kapteyn (cube of 

 10 parsecs side), or q^ = 10— ^^ in astronomical units. We then lind 

 R = 9- 10'\ The total mass then becomes 1/ = 7 • 10'', and con- 

 sequently the galactic system would only represent an entirely 

 negligible portion of the total world-matter. 



It appears probable for many different reasons that outside our 

 galactic system there are many more similar systems, whose mutual 

 distances are large compared with their dimensions. If we take for 

 the average mutual distance 10'" astronomical units, then an elliptical 

 space with /^ = 9 10'' could contain 7 -JO' galactic systems, of 

 which of course only a small number are known to us by direct 

 observation. If, however, they all actually existed, and their average 

 mass were the same as of our own galaxy, then their combined 

 mass would be about 2*10'*, and consequently only one three- 

 thousandth part of the world-matter would be condensed to "ordinary" 

 matter. It is very well possible to construct a world in which the 

 Avhole of the world-matter would, or at least could, be thus condensed. 

 We must then for q^ take the density not within the galactic 

 system, but the average density over a unit of volume which is 

 large compared with the mutual distances of the galactic systems. 

 With the numerical data adopted above, this leads to /? = 5 • 10", 

 and there would then be more than a billion galactic systems. 



All this of course is very vague and hypothetical. Observation 

 only gives us certainty about the existence of our own galactic 

 system, and probability about some hundreds more. All beyond this 

 is extrapolation. 



9. We now come to the case that there is gravitation, which is 

 produced by "ordinary" matter, with the density q^. I will consider 

 the field produced by a small sphere at the origin of the system 

 of coordinates, which I will call the "sun". Its radius is r. 



In the system ^4 the world-matter has thus everywhere the 

 constant density q„, except for values of r which are smaller than 



1) Communicated by Prof. Kapteyn. 



