114 The Correlation Coefficient of a Polychoric Table 
Divided at 11. 
c^ll 
c^ll 
T>Pll 
~ o^ll 
~ o^ll 
bPn 
~ a^ll 
^ a^*ll 
Divided at 21. 
d-f*21 
dP-21 
c^21 
bP 21 
iPzi 
~~ a^21 
bP 21 
6^21 
~ a-P21 
Now if we superpose these two schemes upon an enneachoric table with a 
frequency in each cell, each cell will then contain the P coefficient and frequency 
of each term of the expansion of S-,-,.n-, fwith the omission of the factor ^1 thus 
(~~ d-Pll) dP 2l) ^11 
(c-Pll) (~ dP^l) ''21 
icPn) (cAi) n,. 
(&-P11) (6-P2l) '^12 
(~ aP 11) (b-f*2l) %i 
("~ 0^11) ("~ a^2l) ^32 
(b-fll) (6-f*2l) '^IS 
{~ aPn) ibPzi) ^23 
{~ aPll) dPil) ^33 
When 11 coincides with 21, R becomes = 1 and the mean product degenerates 
into the square of the standard deviation. 
This may be summarised in the following table in which the letter, a, h, etc. 
gives the suffix and the sign gives the sign of the P required. 
Pl2 
-P22 
-d 
-d 
-d 
-d 
%2 
+ b 
-d 
+ b 
-d 
Ml3 
+ b 
+ b 
+ b 
+ b 
«21 
+ c 
+ c 
-d 
-d 
W22 
-a 
+ c 
+ b 
-d 
- a 
-a 
+ b 
+ b 
W31 
+ c 
+ c 
+ c 
+ c 
«32 
- a 
+ c 
- a 
+ c 
«33 
-a 
-a 
-a 
- a 
Thus the coefficient of in *S'i2.2i is (+ 0^*12) (~ a-P2i)- 
This table is sufficient for a polychoric table of any size since any two cross 
points st . s't' in the table, with the planes through them divide it into nine portions 
or groups of cells, each of which is represented by one of the above cells. 
