Series to Frequency Distributions in Space* 

 For all three forms of solutions we calculate 



<* = \/p2Q, d = V / Po2» 



p = Pii/™', 



\ = <Tp l2 la'p 2l (of same sign as />), 



393 





pt 



n is determined by the methods of §7, 8, and 9 by the group 

 of equations A, B, C, respectively, 



A = P3o 2 lp2o\ Pi = PmVpw, 



02 = P*0lP20\ @2 = iW/Wj 



»_ 7 (ft^-ft , +i)-(ft--ft+i) 



6 " 7 (3 A' - 2ft' + 6) - (3 A - 2 A + 6) 



A = PsolpsoPso, fo' = Pto/p&tPoa, 

 2 _ (4ft- 10A + 6A + 2)(4A' - 10 A' + 6A' + 2) K B ) 



"i 



i/*-*u(e+i)0H-i), 



n 2 {0-<j>) + >i(4-4<9) +4(9-4 = 



(C) 



l/?(W)=4 + 0(n-2)7(n-l) 



r=«/(f+l)i r'^Vh + l). 



c = 



<s/n-l 



<r \ > n 



\/){n-r)q(l-q) 



y/r\n-r')q{l-q) 



Equations (C) will in general give the best value of n 

 when they are applicable, since the errors of the lower 

 moments are less, but they may fail to give a real value of n. 

 This is due to the fact that it is not always possible to fit an 

 arbitrary distribution with a hypergeometric series on the 

 assumption <r'= 1. 



(A) and (B) on the other hand always give a real n, for 

 right-hand side of (B) must always be positive, as the 

 equation is only used for symmetrical distributions. 



