104 The Correlation Coefficient of a Polychoric Table 
.: ^ (drdh) = - |(l - 'l^) Pst - B {m., - mjj (48). 
Similarly ^ {drdl-) = - ^ _ P,, - A (m,. - w,,)\^ (49). 
Equation (37) then becomes 
0 = iV (- liHA - ^) a + A' (- MiB - ^) * idlif 
Z' — r7> 
- - r=V2 * (^^'^^''■) + -x^ (^^^'^^^'0 (50). 
.*. Substituting the values of the ^'s, and transposing we find 
1 - r2 r ^ 1 - j V i\7 N^H'' 
N ) 
- N (kKB + r I) 
7)1., 1 
„li — rh I f/, TOs-X n T. / 
+ 2y . — — (51) 
When r = 0 and = — /;,, /vi = — /''a, x reduces to NHK, etc., as above and the 
equation reduces to 
-™™„,=^'|;..(i-1--)-5^(i-1^) m 
If we take this equation for each of the four points 11, 12, 21, 22 and combine 
them according to the scheme = ~~ '"12 ~ '"21 + '^22 ^.s before, the left 
member of the equation becomes 
- NHK {h^l\ - li^l-^ - k;,Jc, + h.l'^) t^.' = - NHK {h^k\ + Ji^h, + h^k^ + Jt^kj) a/ 
= - iNHKhik^a,^ (53). 
In a similar manner the right side reduces to 
( \ k-, ( m., \ 
Yh, ('^- + N ^'-j + NK, + -N "-j 
which may be further simplified as follows 
