534 
Miscellanea 
II. Note on the Surface of Constant Association. 
By KARL PEARSON, F.R.S. 
In the recent memoir by Dr Heron and myself a reference was made to the future publication 
of this note on the surface of constant association, Biometrika, Vol. ix, p. 315. The integral 
equation to the general surface was given, and also the particular form it took, still very com- 
plicated, for the simple case of total or 'marginal' frecpiencies being normal distributions. It was 
considered worth while to fully analyse one case of such a surface, namely that of $=06 for 
Gaussian marginal frequencies. The numerical calculations were carried out by Miss Julia 
Bell, M.A., and from her ordinates of the sections the sections were plotted by Mr H. E. Soper, 
M.A. with the aid of a Coradi coordinatograph. By interpolation when needful Mr Soper 
constructed the isoplethes of this surface of constant association, and also an excellent card 
model, for comparison with the model of a normal surface. The chief features of this surface 
were referred to in the paper just cited. Although the marginal frequencies are symmetrical, 
the cross sections are skew-curves, the skewness increasing from zero for the central section 
to "16 for the y-array when x=l'5<r x and to '20 for the ?/-array when x = 3'5<r x . 
Diagram I gives the series of sections on one side of the mid-section x=0 up to x = 3'5cr x 
by intervals of 0 , 25a- x ; the same sections are repeated in inverse order on the other side of 
the central section x = 0. It will be seen at once from the indicated means how skew the 
sections are. 
Diagram II gives the isoplethes or contour-lines of equal frequency. They are approximately 
but not accurately ellipses with common principal axial directions, but they are very far indeed 
from being similar ellipses. In the contour corresponding to z = j$ of its maximum value, the 
major axis is considerably more than twice the minor axis of the oval ; in the contour corre- 
sponding to 2 = xo> * ne ma j° r ax i s is very much less than twice the minor ; the ovals tend indeed 
to less and less ellipticity. On this diagram are also plotted the regression lines of means and of 
modes. It will be seen that they tend to become parallel to the axis of x, or the regression tends 
to become zero. There is little doubt, that quite apart from normality of the marginal frequen- 
cies, any symmetrical marginal frequency would lead to like results, i.e. the isoplethes would not 
be similar curves, and the regression would be skew, and tend to asymptote to the horizontal. 
Thus the constancy of association would depend for its application on the existence of material 
in which the variatcs would be intimately related near their mean values and cease to have any 
relation towards extreme values. Intervening values would exhibit every variety of relation- 
ship, from the maximum in the neighbourhood of the means to zero value towards the extremes. 
These properties of the surface explain why Q is not even approximately constant, for those 
numerous surfaces of statistical practice in which the regression is approximately constant, 
i.e. linear. 
Plates XXX, I (a) and (b), XXXI, I (c) give photographs of the actual model surface. In Fig. I (a) 
we see the surface ' end on ' to one set of cross-sections and we grasp readily the skewness of the 
cross-sections and its increasing value as we pass from the central section. Fig. I (6) indicates 
by the verticals the regression line of means ; these verticals have their feet on the regression 
line and the eye sees at once by their closer and closer approximation to each other, the deviation 
from linearity. Fig. I (c) gives a diagonal view of the surface of constant association, and under- 
neath it Fig. II (d) has been placed a model of the Gaussian surface* of constant regression for 
comparison ; the angle of deviation was taken roughly about 60°, to give a surface of correlation 
0-5. Except for the regression line and skew- sections in Fig. I, the eye does not distinguish 
very readily in these photographs of Figs. I and II, the fundamental differences of the two 
surfaces which appear so markedly in the isoplethes of Diagram II. 
On the Brill system of interlaced sections. 
