132 
On Polychoric Coefficients of Correlation 
Thus : 
Accordingly 
Tf_, X + 
f_o = \/|5 T, + Vi- 1 Tt_2. 
/ 
h , 1 
X Tf ax ' ^ 
= - 7^ (^/^ ^.^Tt + \/f - 1 %T,_„) (Vii). 
The latter form throws us back on ^^tj which will have to be calculated to 
determine the integral in (vi) for the successive values of t and s. 
On the other hand a table of 
.(viii) 
would be a convenient method of determining the integral and tables of T might 
be easily formed, say up to T,;. 
In this case we may write (vii) : 
f" 
XTtdx = - ^-"^sTt-i (ix). 
1 
We are now in the position to compute all the requisite integrals we need ; if 
we write rigs' foi' the contents of the sth-s'th cell, then on the supposition that the 
surface is normal, has correlation r and follows the actual marginal frequencies, we 
have : 
N 
. k N 
dxdy 
(x), 
''s xzdxdy 
''s-i ''S-1 
N 
+ ... +r"XT^,%.T;+ (xi), 
+ ... +r»%rj,%'Tp' + (xii), 
N J Vi 
+ ...+rP%Tp'^.,rp' + (xiii). 
It is desirable to say a few words about the functions To and T,, which may at 
first present difficulties to the readei'. — t„ clearly stands foi- the integral 
''s 1 _ l i-2 
dx, I.e. 
and is therefore simply ng./N'. 
Similarly — t,' = n.^^'/N. 
