Beatrice M. Cave and Karl Pearson 
341 
where tlje correlations of random pairs of values of the variates, or the product sums, 
are both zero. 
Dr Anderson has further provided us with the values of the standard deviations 
of the successive differences, i.e. 
which represent the ultimate values of 0"^,,,^ and cr^,ny, when we have carried m so 
far that the time effect has been eliminated. 
The new method appears to be one of very great importance, and like many 
new methods it has been developed in a co-operative manner, which is a good 
reason for not entitling it by the name of any single contributor. We prefer to 
term it the Variate Difference Correlation Method. 
With the exception of a few illustrations given by " Student," no numerical 
work on the correlation of the higher differences has yet been attempted. It is 
clear that much numerical work will have to be undertaken before we can feel 
complete confidence in our knowledge of the range and of the limitations of the 
new method. We have yet to ascertain how far in different types of material 
a real stability of difference correlations is ultimately reached, and how far 
various assumptions made in the course of the fundamental demonstration apply 
in dealing practically with actual statistical data. One of the most important 
assumptions made if there be n values of the variates is that arising from the 
reduction in the number of values as we take the means which occur in successive 
differences, and a like assumption is made in the case of standard deviations. 
Thus for example: 
l8{X,) = X, 
n \ 
1 
but =■ S{Xs — Xg+i) = (A\ — Xn) I (n — 1), and will not be sensibly zero, although 
it is assumed to be, unless n be very large. Similar remarks apply to the sums used 
2 re-l 2 
in the standard deviations, i.e. we assume in the proof ^ S (X/) = S (A'-j^i). 
Ultimately with the ?Hth differences we come in the proof to relations of the 
type 
and 
. S (A.) = 'S (A,) 
}t — III 1 n I 
] n — m 1 II 
>S {X/) = -S{X/). 
n — in 1 n i 
Biometrika x 44 
