102 The Correlation Coe^cient of a Polychoric Table 
Xii - Xi2 - X21 + X22 = ^ {H^K^ - HJL, - HJ{, - H.,K.^ 
and the right-hand side of the equation reduces to 
Remembering that 'j^ = «22 this resolves into 
'n,.n.^Y {N -n..) { N - 71.., ) fN-^y fii^y n^. 
N' ) ^ N' ' V^^J [nJ N (^^-"-2^ 
(iV — «.2-) — '^•2) '^2- '^'2 
^5 {W2-'?-2 + (^^ - "2-) «-2 + (^^ - '"-2) 
Hence 
+ (N - {N - n.,)} 
(33). 
N 
This may also be expressed as follows: 
, _ n^. (N - W2. ) W.2 (N - W.2) 
= "22 ("11 + "13 + "'31 + %s) (35) 
1 ] ^ 
AT "22 ('«11 + "13 + »^31 + "33) 
and (T, = — (36). 
When the extreme categories are nearly equal and r is zero = and = K,^ 
and the A'alue for a,.^ becomes infinite. It is necessary then to keep second powers 
of dift'erences to determine a,~. Keeping second order terms we have: 
d-f 
To find etc. we may proceed as follows. 
