Series to Frequency Distributions in Space, 

 while from (38) and (39) 



p 21 2 _ (l-.2qf n-1 (n-2r) 2 r 

 P20 2 Po2~ qQ--q) {n—2y'(n-r)\n—r') 

 pil_ _ (l-2q) 2 n-1 (n-2r') 2 r 



389 



(55) 



(56) 



iW>20 q(l-q) (n-2)* (n-r') 2 (n-r) 



These last four equations involve three more identities as 

 we have four values for the product 



(l - 2 g) '(n-l) 

 2(1-2) (n-2) 2 ' 



The best value of this quantity which plays an important 

 part in the numerical fitting is found from (55) and (56) 

 which involve the body of the table, and we shall take 



(l -2 g ) » (ii-1) _ puftifV* 



q.0— q) («-2) : 



(57) 



«rV 8 (£-l)(i,-l)' 



which is obtained by multiplying equations (55) and (56) 



and writing as is usual a and a' for \/ /> 20 and \/po2- 



If we remember that p=p n /acr' by (48), we can write 

 (57) in the form 



(1-2?)* (n-1) /-,,/',, 



ga- S )(»-2) ! 



0,say . (58) 



PiiXS-l)(v-i) 



These identities necessitate the use of higher moments for 

 the determination of n. 



For this purpose we use (42), (43). 

 Following the usual notation we write 



A - '' :! " 



/3/ 



P'2o I *20 



P<>3 p ' _ P<M 



/'02" P02 



and as in Pearson's paper write 



e=w(»—r)(l— q) 



e =/i(n — r')(l —q) 



~ 2 =?• que 



Zi = € + r<]U 



Zj =6 +r qn 



> 



(59) 



60 



that 



e ??— r' ~ 2 r n—r' 

 = and — = — , 



n — r 



r n — r 



