26 



C. V. L. Charlier 



of the whole duster with the number of stars contained in it. If m and |jl are 

 known we are able to compute n. 



Consider for instance the brightest of all globular clusters a Centauri. Its in- 

 tegrated apparent magnitude may, according to John Herschel, be estimated as 4.5. 

 How many stars of the apparent magnitude 15 would it contain for giving rise to 

 this integrated magnitude? We find n = 16000. 



In reality the stars in a cluster have not all the same apparent magnitude 

 (and hence also not the same absolute magnitude). The relation between h, m and 

 n is therefore somewhat more complicated. 



Let /(m) be the frequency function of the magnitudes. Writing 



-1/^—0.4 m — 2hm 



10 = e , 



where 



b = 0.4605 



we then get as expression ot the total brightness (= X h) of the cluster the formula 

 (17) Yh= C^e'-^'"' f{m)chn, 



— œ 



where C is the same constant as above. 

 As to f{m) we know that 



+ 00 



^/(w) dm 



— cc 



is convergent and we may without hesitation assume that /(m) does not possess, 

 for a finite dominion of m, an infinite number of maxima and minima. According 

 to L. M. 71 the conditions are then fulfilled for developing f{x) into a series of 

 type A. Put 



where N denotes the total number of stars. 



Neglecting the higher characteristics we may use for f{x) the first term in 

 this series. Substituting in (16) and integrating, we get 



or 



= C 1 0~ ~ 



where denotes the mean value of the magnitude of the stars and a the dispersion. 



From this formula we get for the integrated magnitude, jx, of the cluster the 

 expression 



(17*) [J- = — — 2.5 log iV. 



For a = 0 we fall back upon the formula (16). 



