38 



C. V. L. Charlier 



tween the value of x from the second equation with the other values is a check on 

 the calculation of iV from (66*). It is found that an error in N only in a small 

 degree influences the corresponding values of x and y. From x and y we get for 

 (Ä; = \.ly) and W0(= hx) the values : 



7c = -|- 3.108; w?o -f 17.59. 



The values of the same parameters for are: 



iNT = 30 000 000 

 A- 3.119 



W/y = + 20.07 



the three values of x from (65*) are: 



,X =^ -|- 6.434 

 X = \- 6.436 

 X = + 6.434 



of which the second value is a cheek on the calculation of iV. 



In (in the Milky Way) the number (a(««)) of stars of the magnitude m is 

 increasing to m = 20"'. oi, for which value a[m) is a maximum. In (7, the maximum 

 value is reached already for m= 17.6. It is evidently within the power of stellar 

 photography of our days to decide definitively if these conclusions are in agreement 

 with the real state of the universe. The formulai for a(m\ are 



(67) 



, ^, , , 630 000 

 In C • a m = - — c 



(TO- I 7.53)^ 



2jt2 : 



where h = 0.108. 



(67*) 



In a, 



Tab. 6. C, 



30 000 000 _ {>n^2ä <^ 



a{m) — 1= — e 27.2 • 



7f 1/271 



where h — 3 119. 



TO 



A{yn) 



2.0 



0.10 



3.0 



0.68 



4.0 



3.91 



5.0 



20.5 



6.0 



97.2 



7.0 



417 



8.0 



1 682 



9.0 



5 780 



10.0 



18 570 



11.0 



54 200 



12.0 



144 000 



13.0 



348 000 



14.0 



768 000 



150 



1 547 000 



16.0 



2 904 000 



17.0 



4 906 000 



18.0 



7 640 000 



19.0 



11 010 000 



20.0 



14 740 000 



It is remarkable that the values of It in both squares seem 

 to be identical. 



According to (65) the whole number of stars brighter 

 then m is given by the formula 



(65) 



/1(?W.) 



0 



In the following table I give the number of stars in C^, 

 calculated with this formula. 



Between m = 6 and m = 9 the quotient A{m A- 1) : A(m) 

 is about 4.0, agreeing with the observations. This quotient 

 however (in the table) continually decreases. For making a 

 comparison between the formula and the observed values of 

 A[m) the following table was computed: 



