Series to Frequency Distributions in Space. 387 



Patting d = (l + rq)c, d' = (l + r'q)c', we obtain after some 

 reduction 



7^21 



P\2 



P20 



+ : ?mq(nr-2r-2r' + n)+(n-l)-r' + 2r] 

 cc 



-Q.+r'q)qr{l-q)(n-r-r<)=;0, . . (34) 



-H^[g(n/-2/-2r + 7i) + 0i-l)~r + 2r'] 



cc 



-(l+rg)?r'(l-g)(rc-7— r')=0. . . (35) 

 We now substitute the previously obtained values for 



p 2 Q, 'pot, Pn, and after some more reduction obtain 



n-l 



c c cc 



& (,/_ 1) + ^(„_ r _l)= - "•'?(! -g)(l-2g)(n-.--r') 



(iV v y CC 2 v ' ft — 1 



Solving these equations we find 



cVr/ 9 (l-o)(l-2o)(»-2r) 



• (36) 



• (37) 



Pa 



Pl2= — 



(«-l)(»-2) 



cv , -V) J y(l-<y)(l-2r / )(ft-2/-') 

 («_1)(«_2) 



• (38) 



• (39) 



Four additional marginal moments will be required, these 

 can be deduced from Pearson I. c. : 



_ *rq(l^q)(l-2q) (n-r)(n-2r) 

 />;J0_ ( n -i)( n _2) 



t .'*r'q(l-q)(l-2q){n-r')(n^2r') 

 ^ 03 ~ {n-l){n-2) 



(40) 

 (41) 



_ c*q(l-q)(n-r) 

 ^-(^1)^-2)0.-3) 



c*g(l-g)Qi -Q 

 *"-(n-l)(n-2)(n-3) 



r >r(l + 3yl -qv- 2) 



+ n@ql^r6^+l-6r \ (±2) 

 ,+6/ 2 (l-3^1^) ) 



1 



n 2 (l + 3?l- </>•'- 2) 



+ n^T^r'S -/ + 1 - 6r') I (43) 



+ 6r B (l-3?T^) J 



2 C 2 



