1914] on The Stars around the North Pole 133 



23. In consideriDg the luminosities of the stars it will be con- 

 venient to choose the distances 50, 63, 79, lUO, 126, 158, 200, 251, 

 316, 398, 501, 631, 794 parsecs. These distances are in geometrical 

 progression, and are chosen so that if the same star were moved 

 from any one to the next greater it would appear to us half a 

 magnitude fainter. Thus the sun at tliese distances would be of 

 magnitudes 9-0, 9-5, 10-0, 10-5 .... 15-0. 



Next let us suppose a cone drawn with its apex at the sun, em- 

 bracing the area round the North pole, and divided into boxes by 

 spheres of these radii. The volume of each box will be twice as great 

 as the one next to it and nearer the sun. The number of stars in each 

 box is indicated in Fig. 7. Let us consider the four boxes A, B, C, D. 



In volume D = 2 C, C = 2 B, and B = 2 A. 



If D be taken as a standard volume, the number of stars in a box 

 this size placed at A, B, C, D, respectively would be 200, 172, 132, 

 98. Thus the density as we go farther from the sun diminishes. 

 But we must remember now that all these stars seen at S appear to 

 be of 10" "5. Thus the stars in D must be brighter than those in C, 

 those in C than those in B, and those in B than those in A. A con- 

 tains stars of the same absolute magnitude as the sun up to those 

 I magnitude brighter. B are from J to 1 brighter, C are from 1 to 

 1^ magnitude brighter, and D are from 1^ to 2 magnitudes brighter 

 than the sun. 



24. If we may assume that the actual density of the stars is the 

 same in all parts of the space with which we are dealing, we obtain 

 by reasoning of this kind the number of stars ! etween different limits 

 of absolute brightness, and it is found that in a sphere of 100 parsecs 

 radius the numbers are as given in the following table : — 



The results in the second column have been arrived at by con- 

 sidering the faintest stars, those from 10™ "0 to ll^'-O ; those in the 

 last column by taking the stars which appear to be of magnitudes 

 9-0 to 10-0. ' 



