250 REPORTS ON THE STATE OF SCIENCE. 



is the Milky Way itself, which introduces a complication and exag- 

 gerates in an irregular manner the already great star-density. But 

 in the main the concentration cannot possibly be attributed to the 

 Milky Way, which obviously can only be responsible for the star- 

 density in the very limited belt in which it lies. 



There is a very important theorem in stellar distribution which 

 seems to have been known for a long while, but I do not know who 

 originated it. If in any direction the stars are uniformly distributed, 

 as far as the instruments will penetrate, then the number of stars of 

 one magnitude is four times the number a magnitude brighter (more 

 precisely, the ratio is 3 98). This allows for the fact that individual 

 stars are of all degrees of intrinsic brightness ; it only requires that the 

 mixture of big and little stars should be in the same proportion every- 

 where. If this holds, and the stars extend uniformly for an unlimited 

 distance, then there will be four times as many eighth as seventh 

 magnitude stars, and so on. 



Now it is well known that this star ralio, as it is called, falls far 

 short of its theoretical value. I give a table of the star ratio based on 



Slur Rntio". 



Kapteyn's work. 5 Even in the galactic plane it is much below the 

 theoretical limit, and towards the poles it is even smaller. An obvious 

 interpretation is that the star-density falls off at some distance from the 

 solar system, and that the falling-off is more pronounced at the galactic 

 poles than near the equator; this leads to the conception of the bun- 

 shaped system already described. To give some idea of the extent 

 and scale of the system the following may serve : A sphere of radius 

 corresponding to a parallax 0"'2 was found to contain eighteen stars. 

 Within a sphere of ten times the radius (parallax 0"'02) there is no 

 appreciable falling-off in density; but now multiply by ten again 

 (parallax 0"'002), and towards the galactic poles you will reach a region 

 almost void of stars, whilst in the galactic plane the falling-off in 

 density will probably be appreciable, unless, indeed, you have reached 

 the star-clouds of the Milky W T ay itself. 



These star-magnitude statistics at present suffer very much from 

 the absence of reliable standards of magnitude for the faint stars. Thus 

 the figures for the star ratio given in the table are open to considerable 

 doubt from that cause. We are, however, confident that that diffi- 

 culty will soon be removed, and the data will be vastly improved. The 

 work of Professor Schwarzschild on the brighter stars, and Professor 



5 Oroniiigen Publications, Ne. 18. 



