110 SYDNEY CHAPMAN_, B.A.^ D.SC, F.R.A.S., ON 



fainter and yet fainter stars, their number increases very much, 

 and at a hasty glance it might ahuost seem tliat their number 

 was really infinite, that they increase without end. But if the 

 table be examined more critically, it will be seen that the total 

 number of stars down to the 6th magnitude, is nearly four 

 times that down to the 5th ; but that the total number down to 

 the 10th 'magnitude, is not quite three times that to the 9th ; 

 and the number to the 17th not even twice the number down 

 to the IGth ; so that though the number of stars down to any 

 particular magnitude is always larger than the number down to 

 the preceding magnitude, yet the ratio of the increase is 

 continually diminishing. The number of stars of a given 

 magnitude does not increase in so high a geometrical ratio for 

 the fainter stars as for the brighter ones ; and a mathematical 

 examination of the actual numbers in the table shows that two 

 conclusions can be drawn with regard to the whole number of 

 stars, seen and unseen. 



In the first place it appears from this examination that there 

 is a total number of the stars ; that is to say that the number 

 of the stars is not infinite. As we go from the number of stars 

 of any one given magnitude to the number of the next fainter 

 magnitude, we are dealing with a series which does not tend to 

 increase without limit : the ratio of increase continually 

 diminishes, and therefore a point will be reached beyond which 

 the actual number of stars of any particular magnitude will no 

 longer be greater than the number of the preceding magnitude. 

 The series becomes a " convergent " one, and the total number 

 of stars must therefore approach a limit ; in other words it is 

 finite ; an extremely large number as we shall see, but quite a 

 finite one. The stars therefore vary in l^rightness from Sirius, 

 the brightest star of which we know, down to the 17th magni- 

 tude, the limit for the Franklin- Adams photographs ; and still 

 fainter than any limit which at present we can possibly reach ; 

 indeed very much fainter. In fact there are possibly stars of 

 almost all conceivable degrees of faintness, but their total 

 number is limited, and this conclusion is enforced upon us, and 

 generally accepted, on other grounds beside those indicated 

 above. 



In the second place, the series shown in the table and 

 derived from these counts of the Franklin-Adams plates gives 

 us an indication of the limit of magnitude to which we should 

 have to penetrate to secure half the total number of stars. As 

 we have seen, the plates themselves carry us down to the 17th 

 mao-nitude with the images of some 55,000,000 of stars. This 



