Miscellanea 
193 
Whence using (xiii), (xvi), (xix), (xi), (x) and (xvii) we have : 
o-„ o-„ , ,R„ „ , , = u„q u„<„, ( + ) (xxiii) 
Taking p=jo' in (xxii) and using (xiv), (xviii), (xvii), (xi), (x) and (xii) we find : 
o-u o-„ „ , = Mp,!wf— - + - — ) (xxiv) 
( 11 n f 3 \ 
Similarly : o-„ tr,. , R„ „ , — ^- H — ) (xxv) 
*' "PQ "p'Q "pq"p'q Mjj'Jlg ^ 
Turning back to (vi) and taking differentials, we have : 
2<^8^ = <Sp,(5Mp,). 
Square, sum and divide by the number of random samples, and we find : 
r 0 Pq\ "pq' ' «p3 "p'q' "pq^^p'q" 
+ 22, (o-,, a-,. , R„ „ , ) 
+ 22,-i(a'„ (T,, , i?,, „ ,), 
• I a \ Upqf "'^iq^'pq'" 
where Sj denotes a summation for all unlike values of p, p', q and q' ; 2^ for all like values of q 
and unlike of p and p' ; 23 for all like values of p, unlike of q and q'. 
Substituting from (xxi), (xxiii), (xxiv) and (xxv) we have : 
+ 22iL„iV9'f— "+-^^- 
+ 222 [Vp'q + ^ - 
i * KllpTlq Up, Tig nqjj 
+ 223 {upn^^pq' (^ + ^ - -^1 (xxvi) 
We will consider the parts of this in order. Let \jfpq be taken = ^ (^ipq - -^-^ = contri- 
bution from any constituent to the mean contingency. Then 
(})\q = JV^^'^jJllp 7lq (XX Vii) 
Hence JV^Spq f ^) (xxviii) 
remembering that Spq {\j/pq)=0 and »S'p, (?ipW,) = iV^. Next consider 
^ [^^pq\i22 ( ^''"i^''i'' <i \ — g { ^P (^^pg^ ^ '^P (^pq) l 
""KnqJ ^\ Uq J "[ Uq J 
Biometrika v 25 
