18 
PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS 
of points determined as the means of arrays of x for given values of y, we have once 
and for all freed ourselves from the difficulties attendant upon assuming normal 
frequency. We become indifferent to the deviations from that law, merely observing 
how closely or not our means of arrays fall on a line. When we are not given arrays 
but gross grouping under certain divisions, we have seen that the “ transfer” is also 
a physical quantity of a significance independent of normality. We want accordingly 
to take a function which vanishes with the transfer, and does not diverge widely 
from the correlation in cases that we can test. Here the correlation is not taken as 
something peculiar to normal distributions, but something significant for all distribu¬ 
tions whatever. Such a function of a suitable kind appears to be given by Q 5 . 
§ 6. On the “ Excess ” and its Relation to Correlation and Relative Variability. 
There is another method of dealing with the correlation of characters for which 
we cannot directly discover a quantitative scale which deserves consideration. It 
is capable of fairly wide application, but, unlike the methods previously discussed, it 
requires the data to be collected in a special manner. It has the advantage of not 
applying only to the normal surface of frequency, but to any surface which can be 
converted into a surface of revolution by a slide and two stretches. 
It is well known that not only the normal curve but the normal surface has a 
type form from which all others can be deduced by stretching or stretching and 
sliding. Thus in 1895 the Cambridge Instrument Company made for the instrument 
room at University College, London, a “ biprojector,” an instrument for giving 
arbitrary stretches in two directions at right angles to any curve. In this manner 
by the use of type-templates we were able to draw a variety of curves with arbi¬ 
trary parameters, e.g., all ellipses from one circle, parabolas from one parabola, 
normal curves from one normal curve template. Somewhat later Mr. G. U. Yule 
commenced a model of a normal frequency surface on the Brill system of inter¬ 
laced curves. This, by the variable amount of slide given to its two rectangular 
systems of normal curves, illustrated the changes from zero to perfect correlation. 
This model was exhibited at a College soiree in June, 1897. Geometrically this 
property has been taken by Mr. W. F. Sheppard as the basis of his valuable paper 
on correlation in the ‘ Phil. Trans.,’ A, vol. 192, pp. 101-167. It is a slight addition 
to, and modification of, His results that I propose to consider in this section. 
The equation to the normal frequency surface is, as we have seen in § 1, 
z 
n r 
2^yT=? ex P‘- 1“ 
2 rxy y~\ 1 1 
o‘ 1 <r 3 ^ <riy 1- riJ 
Now write x/(cr l \/l—r-) = x', y/cr 2 = y'. This is merely giving the surface two 
uniform stretches (or squeezes) parallel to the coordinate axes. We have for the 
frequency of pairs lying between x, x -f- Sx, and y, 8 Sy, 
