192 
Miscellanea 
%%,^„,n,, = --7y-> (Xll) 
U 11 , , = — , (Xlll) 
0-n O-,, „ , =_^W3^ (xiv) 
0-n <r„ , Rn n . = - , (xv) 
"■pq "p'o ^^pq^p'q N 
0"n„ 0"r» 
o-„ (T,, , R,, n , = — ^ifT^ , (xvii) 
°'''p"'''pq - "p"pq 
^^q ^'^pq "(J 
where throughout and p' and y and ^' denote different columns and rows, and 7? is a correlation 
coefficient between subscriiat frequencies. Taking logarithmic differentials of (v), i.e. treating 
the deviations of random sampling as of differential order as is usually done, 
Swp, ^ ^biipq _bnp_bn^ ^^^^ 
Then squaring, .suniming and dividing by the number of random samples : 
(T^U t^^'n f^ii 0"h (7"« Rn « (T ») (r« Rn n 0"« R*i n 
PQ _ PQ _^ jy _| -?^2 " p"g_ ^ "fig "/) _ ^ »g "pg"<; _ 
'^^pq '^pq '^^p^ '^^q^ '^^pf^q "^^pq^p ^^pq'^^q 
Whence using (vii), (viii), (ix), (x), (xviii) and (xix) we find 
—«'=,( 1 ^ ) (xxi) 
Vhq lip 7lg llpTlJ 
"pq : 
We have next to find the correlation of deviations in Upg and ?<p.g. due to random sampling. 
Three cases can occur according as p and q are equal or unequal to p' and q' resiDCctively. We 
have by multiplying (xx) by a similar form : 
°""pg "^"p'q' ^"pq"p'q' _ ^ "^"p q '^'"■p'q' ^"pg'V'q' _ g °^"p' ^"pg"?/ 
'ff'pq'i^p'q' '^Ipq'^p'q' ^^pg^^p' 
'ilpq 7lq' Hp Hp'q' 
Tip Tip' Tip Tlq' 
_ 2 ^^'q^'^p'q'^^q^p'q ' ^ ^^q ^^p' ^^^q»p' 
llq Tlp'q' HqTlp' 
_^ %%--^»g"g- (xxii) 
