108 The Correlation Coefficient of a Poh/cliorie Table 
~ A ~7\. ^ {C^st^^st ( — + ^0011 -^llsi + ^OOlzRlZ-st 
»00si 
^ - ^0000 I V cr,t I 
_ ^ ^''.sf-^onno _ ^onuo H'W 
and since 26',^ = 1 a,^ = (72). 
'^0000 ~ ^ 
§7. Comparison of Polychoric and Enneachoric Coefficients of. 
Correlation. 
We may now compare the polychoric coefficient (/'j,) with the enneachoric 
coefficient (rj previously found. 
Since S (C,^) = 1, 
+ Cj2r32+ (73), 
••• = Cnf^r^i + C'^^Jri, + ... = - C'n — - C/i^ (74), 
Xll Xl2 
c c 
= — ^ (y4iicZ?»i. + B-^^dm.-^ — dm-^i) -\ — {A-i^dnii. + B^^^m-z — ^^'"12) + etc. 
Xu X12 
(75). 
Transposing and rearranging we find 
— dm-,, + — (^m,, + ... 
Xll X12 
C C r 
— (^ii(/h?i. H- B-^idm.-^ + Xll ^''11) + ^(^i2'^"h.+ -5i2f^"? -2+Xi2<^*'i2) + etc. 
Xll X12 
Xll 
(76), 
where = iV j j z {x, y, r) dx dy = m^f , 
00 —00 
"hi + »«12 + - = ^ (1^0 A + ^10) + (iTo A + ^12) + ... 
Xll X12 Xll X12 
(77), 
using 'S'si as in equation (61), page 106, which is an equation of which is a root. 
If in this equation w^e put C\i — Xii> Ci?, = ~ Xi2' ^'21 = ~ X21' ^22 = X22; the 
remaining C"s = 0 we have the equation for the enneachoric coefficient and 
now appears to be the weighted mean of four tetrachoric r's, 
r = Xll^'ll - Xl2yi2 - X2iy21 + X22?'ii2 
Xll - X12 - X21 + X22 
