Karl Pearson and David Heron 
315 
By the equation in the second line of p. 184, we have 
(l-%)(P+g) + iV r + V|^-(l - x)(j>+g)l 2 + 4%(l-%)Fi 
Whence by straightforward but somewhat laborious differentiation we find 
f x f v x{N*-N(l- x ){p + q)+l(l-X)P<l\ 
z = - — (i). 
[\N-(l- X ){p + q)] i + HO--X)P ( lY 
This is the equation to the surface of constant Q. As soon as we know the 
nature of the marginal frequencies, i.e. the values of p and q, we can find the form 
of the surface. The above equation is somewhat simplified if we refer p and q 
to the medians of their frequency-distributions, i.e. write p = hN — a, q= — /3. 
In this case 
5gfxH*'(i + x)-«(x-i>«fl) 
If the marginal frequencies are Gaussian, 
1 T" 1 V" 
2f rx - ± ±s N fy - s —>. ? 
a= . e ' 2<r i"d:r, /3= . e 2<r2 dy, 
\/27rovo \/27ra.,-io 
it a., 
It is therefore possible by aid of Sheppard's Tables to construct the contour 
lines of the surface of constant Q for this relatively simple case. But the surface 
is far from simple and its complex equation seems to indicate that association as 
measured by Q is of a very arbitrary character. We have constructed the surface 
of constant association for the special case of Gaussian marginal frequencies, when 
Q = '6. The photograph of the surface, the regression lines and the contours will 
be published on another occasion. It suffices here to note : (i) that the arrays are 
heteroscedastic, varying from homoscedasticity of the mid-section to a skewness of 
•16 when x\<r x = 1'5 and to a skewness of - 20 when x/cr 1 = 3'5. (ii) The regression 
line is most markedly skew, in shape like a Galton ogive, so that there is a 
maximum of regression, and therefore correlation, at the centre of the surface, 
while the regression and therefore correlation reduce to zero as we move outwards. 
No frequency surfaces in actual practice exhibit, as far as we are aware, these 
features demanded by constant Yulean association. 
40—2 
