The distribution of the stars to the eleventh magnitude 



51 



This equation shows, that E{m) is proportional to the inverse relative density. 

 For the value iiIq— 11,0 for instance, we simply have the reduction equal to the 

 inverse relative density: 



m,= ll,0, 



As the quantity J- may be calculated for each zone, we may accordingly take 

 the value of the limiting magnitude directly from tab. VII. For any other value 

 of the factor JÎ(>»o) directly given. The quantity is not known and 

 is indeed impossible to determine by observations. However, the value does not lie 

 far from eleven as the table of the limiting magnitudes allows to be supposed. In 

 order to see the influence of a variation of I make four hypothesis, putting 



= 10,5; 11,0; 11,5; 12,0 

 and calculate by lielp of tab. VI and tab. VII the magnitudes of the seven zones, 

 making use of the same division into groups according to the star density as before. 

 (In calculating the mean value of ^ I have at first inverted all the values of d.). 

 The result is given in the following series. 

















1 



10,5 



l(),'i5 



10,78 



10,64 



10,-16 



10,86 



10,21 



10,02 



11,0 



11, 2Ü 



11,28 



11,13 



10,96 



10,86 



10,70 



10,47 



11,5 



11,77 



1 1,79 



11, C5 



11,46 



11,30 



11,20 



10, 90 



12,0 



l'i,30 



12,33 



V2,U 



11,96 



11,85 



11,67 



11,44 



These series are easily discussed. The seventh zone contains only 2 per cent 

 of the observations and is therefore neglected. The magnitudes of the other 

 zones show: 



1. The magnitude of the first and second zone is the same. 



2. The decrease of the limiting magnitude is proportional to the distance 

 from the centre. 



3. The decrease from one zone to the next. Am, is slightly increasing with the 

 supposed value of as the following series shows: 



10,5 11,0 11,5 12,0 



àni 0,1525 0,1625 0,1650 0,1775. Mean value 0,160 ± 0,004 

 According to 1 the first and second zone are combined to a single one, z^, 



and the next is denoted with etc. 



According to 2 and 3 we have, denoting the central magnitude with m^^ and 



that of the zone z„ with m„ as a result of this discussion and with an accuracy sufficient 



for the present purpose, the value of the limiting magnitude given in the following 



expression : 



wi„ — — 0,16 . n. 



