CHAPTER VIII. 



The mean errors and the effect of a varying ft, on the 

 positions of the vertices and the values 

 of the characteristics. 



24. We shall in this chapter at some length derive the method to find the 

 mean errors of the characteristics. It is then necessary first to find the mean errors 

 of the characteristics of the motion as projected on a square. This will be the 

 care of this paragraph, which also may be regarded as a supplement to the mathe- 

 matical theory of the correlation function of the A-type as developed in chapter I. 



The problem to find the mean errors of the characteristics of a correlation 

 surface has, though not yet published, been considered by Charlier. By his kind- 

 ness I have had access to a manuscript in which he gives the general formulae 

 necessary for solving the problem, and the outlines of the treatment of the higher 

 characteristics. 



Denoting by s(x) the mean error of the quantity x, Charlier gives the 

 following general formula for the mean errors of the moments of the correlation 

 surface (N = the number of stars) : 



(69) 



iNi, m ) = -)=. VN»,*m - N r > m . 



From this formula it follows immediately that 



= -JM/2 + 24ß 40 



(70) 



'22 



In the following we will confine ourselves to the case that the distribution is 

 nearly normal. We then have 



