386 
Supplementary Tables for Tetrachoric Groupings 
It has been shown by Elderton* that 
i - } r r*-*-^***-^ **** 
may be transformed into 
where t — - . and this form is utilised to compute d as a check for the value 
v 1 — r 2 
of r obtained by the usual method of solution. Now -r= I e 2 ' cLY and 
— = e~^ r are known quantities, being the ^(1 + a) and z of Sheppard's Tables^. 
V27T 
Putting ^ (1 + a) = / for convenience, we now have = J zldy and this form was 
used for constructing the tables. 
The work was carried out in columns ; the first column contained the values of 
y or k rising from zero by intervals of "1, while the second column contained the 
corresponding values of z as found from Sheppard's Tables. These two columns 
remained constant during the whole of the work. The remaining columns were 
grouped in threes and contained the values of t, I and zl, the headings having 
suffixes attached showing the value of h to which they applied. The values 
of zldy were now obtained by a quadrature formula, first using all the values of 
J k 
zl in the column and then striking out each in turn from the top of the column 
downwards. Considerable difficulty was found in choosing a suitable quadrature 
formula and after many trials Weddle's Rule was adopted as being the most 
suitable j. The table for r = 1 was formed directly from Sheppard's Tables. 
Checking. 
Since } f' \\-\'^ +yi - 2r ^My 
N 2tt\/1 -r*Jh Jk 
is perfectly symmetrica] in x and y, it is obvious that the value of djN is unaltered 
when the values of h and k are interchanged. Consequently (excluding the case 
h = k) each value of djN in the tables occurs twice and by the above method of 
construction each of the two values is obtained independently ; it will also be 
noticed that in the final quadrature, the number of ordinates employed will also 
* Frequency Curves and Correlation. Layton. London. 
t New Tables of the Probability Integral. W. F. Sheppard. Biometrika, Vol. n. pp. 174 ff. 
X It was always possible to make the ordinates number a multiple of six, as required by Weddle's 
Rule, by the simple process of adding sensibly zero ordinates at the asymptotic tail of the curve. 
