404 ASTROMETRY 



For the limits of parallax I chose numbers increasing in the con- 

 stant ratio of 1 to 1.585. They are such that, if all the stars had the 

 same luminosity their apparent brightness would diminish by just 

 one magnitude as we pass from one shell to the next more distant 

 one. 



Some of the earlier investigators have started from the hypothesis 

 of equal luminosity of all the stars. In their theory, if the first shell 

 contains the stars of the first magnitude, all the stars of the second 

 magnitude will be contained in the second shell, those of the third 

 in the third, and so on. 



If we treat the stars of the other magnitudes and proper motions 

 in the same way, we shall get what has been represented in Fig. 2. 



We there see that of all the stars of the sixth magnitude, 614 will 

 finally find their place in the 4th shell (x =0"016 to 7r-0"025), 833 

 in the fifth, 901 in the sixth, 771 in the seventh, etc. 



It will be remarked how widely this arrangement differs from what 

 it is in the just quoted theory which places all the stars of any one 

 apparent magnitude in the same shell. 



All the stars of the same apparent magnitude have been put down 

 in the same sector; we have of course to imagine them distributed 

 through the whole of the shell, mixed with those of other apparent 

 magnitudes, the numbers of which have been inscribed in other sectors. 

 No more than seven shells could well be shown in the figure. In 

 the actual computations their number is of course increased and the 

 stars of the second, third . . . seventh, eighth, ninth magnitude 

 were duly taken into account. 



The complete figure would thus show the distribution in space of 

 the stars of any apparent magnitude between the second and the 

 ninth. 



In principle no other hypothesis was introduced beyond our funda- 

 mental hypothesis, which supplied the mean parallaxes. It is true 

 that the frequency-law introduced is still somewhat unsatisfactory 

 and will remain so for some time to come. The objection, however, 

 is not fundamental. As remarked already there is nothing to hinder 

 us from finding out its true form as soon as we have a sufficient num- 

 ber of accurate parallaxes at our disposal. 



In order to derive further results we shall now introduce a new 

 hypothesis, viz., that light suffers no absorption in passing through 

 space. We shall presently have to discuss in how far the results are 

 changed if this hypothesis is dropped. 



Admitting that space is perfectly transparent we can at once 

 derive the absolute magnitude for every star in our figure. Abso- 

 lute magnitude of a star we shall call the apparent magnitude this 

 star would show, were it placed at unity of distance. I shall here take 

 unity of distance to correspond with a parallax of 0"01. The stars 



