Karl Pearson and Egon S. Pearson 
131 
to determine hg. and ^.j-. It is then often simpler to work directly with (iv) rather 
than interpolate into the tabled values of the functions. 
In an earlier paper* dealing with the tetrachoric functions one of us has shown 
that if 
z = 
1_,.2 
27rV(l -r^) 
be the equation to a normal correlation surface the variates being measured in the 
standard deviations as units, then 
ZIN = Tit/ + 2?-T2To' + 8r=T,,T;/ +...+(<+ 1) r*Tj+,T',+i + (v), 
where Tt = Tf (.^■) and t/ = t, (y). 
Now in order to proceed further it is needful to xdetermine the following 
integrals : 
I Ttdx, \ XTfClx. 
We can determine these by using (iii) after in the second case integrating by 
parts. We have : 
f'. , 1 [h / dV-' 1 
Ttdx = -p= 
1 
{ dx) 
dxj V27r 
-r-^e 
\/27r 
dx 
1 
Again ; 
XTfdx = 
4^{-x) 
.(vi). 
V27r 
e ■'^■^ dx 
dV-- 
dx) ^2 
1 
e - 
+ 
jLj"- (_ 
-j^rt--iX I I 
yt A 'is-i V t J /is-, 
dx) V27r 
T^_i ax 
_1 1 
But by (iv) : 
1 
7t 
T^_l X + 
Pt 
.(vi) bis. 
where 
* Phil. Tram. Vol. 195 a, p. 4, Equation (xiv), with a slight change of notation. In that paper, 
1 -hx-i 
ji+i 
, , is written for r,,, , and -X= e -^"^ -^^' ^ — for r' 
9—2 
