22 



Sven Wicksell 



(45*) 



(4ô* 



1 = V 2 1 + V 2 0 Vo + 2V 11 X 0 + 4 



2 = V l 2 + V 0 2 X 0 + 2V 1 1^0+^0 ?/l 



3 = V 0 3 + 3V 0 2 Po + ?/o 



V 4 0 V 4 0 ~H ^ V 3 0 ^0 



+ 6 V 2ü 35*0 + < 







V 3 1 = V 31 + 3V 21 33 0 



+ v 3o2/« + 3v n 



x\ + 3v, 0 .x 0 y 0 



+ < y u 



V 2 2 = V 2 2 + 2V 2lV0 





i ^ u o ~r v o 2 ^ o 



+ v 2ü? / 0 



v 'i3 = v i3 + 3v i2 y 0 



+ v 03 *ü + 3v n 



y\ -f 3v U2 a; o //o 



+ *o tfi 



v'o4 = V 04 + 4V 03 Vo 



4- 6v o2 y\ + y\ 







and mutatis mutandis the saine equations for 2V^. 



On the other side the moments about the mean may be expressed in the 

 moments about the origin by similar equations. 



1 1 . From equations (44) we have 



u + x n =± r (U+ X 0 ) 



v + v 9 *-f{v+ r 0 ). " 



Assuming the linear velocities of the stars to be independent of the distance 

 we thus have 



or putting 



we write 



(46) •,-• = »,.+,■ W< 



Now Charlier uses the notations 



a(m) dm — the number of stars having their apparent magnitudes between 

 m 4- Va dm and m — V* dm, 



¥> 0 (M) dM = the number of stars having their absolute magnitudes between 

 M + Va dM and M— 1 /* dM, 



and putting 



. —by 



~~~ b = 0.2/mod 



y — — 5 log r 



\(y) = the number of stars having y between y -4- Va dy and y — '/a dy. 

 Then evidently as 



M = m + y 



