Egon S. Pearson 
37 
V. On Methods of Reduction. 
(a) Variate Difference Correlation. 
It will become evident in the detailed discussion of the results of the experi- 
ments, that a considerable part of the correlation of the successive judgments 
is due to a secular change with time, occurring from series to series, and in the 
case of the Trisections, to a sessional change as well occurring within the series ; 
I therefore propose to consider at this point how far the Variate Difference Corre- 
lation Method is applicable in this type of problem, and to do this will approach 
the matter from a slightly more general point of view than that of " Student " in 
Biometrika, Vol. x. p. 179. 
Suppose that x and y are the two variables to be correlated, with corresponding 
values 
Vi, }h, ■■■ yt, ••• Vv 
and that we may express Xt and yt in the form 
yt = F,{t)+Yu 
where Fi (t) and F., (t) are polynomials of degree n in t, the unit of t being the 
interval of time or space between the successive values of the variates, which is 
supposed equal and constant ; Xt and Yt are independent of the secular or sessional 
change represented by F^ and Fo. 
Let us now obtain a general expression for 
(1) 'a A ijt or nR, the correlation of the Hth forward differences of Xt and yt. 
(2) 'X^Y,; A,r, or „ „ „ Xt „ Yt. 
Now 
n ' 
AnXt = (1 - e)" Xn+t = x„+t - nxn+t-i . . . (- 1)* x.^+t-s • • • (- 1)'' Xf . .(xiii), 
where the operator e is defined by e^Xt — Xts, etc. 
Further we must assume that 
V 
(a) "Z Xt+h = constant for all values of h small compared with v, 
= 0, by suitable choice of origin, 
V 
S yt+h = o, 
from which it follows that S A„a:t_^/, = 0 — S ^nVt+i,, 
V 
(h) 2 {x-f+h) = constant = va^' for all values of // small compared with v, 
V 
S iy't+k) — '^'^y^ " » " » >' 
