Egon S. Pearson 
39 
substituted for xyp xP ^i-nd yp. But as F^{t) and F.,{t) are polynomials of degree 
n in t, we know that 
AnXt = AiiXt +• constant [ 
Anijt = Yt + constant] ' 
and therefore 
that is to say we may equate nR to an expression similar to that on the right hand 
side of (xvii) above, except that the correlation coefficients of the residuals, namely : 
xyP' xP ^-ricl yP ai'e to be substituted for xyP, xP and ,ip. 
Now in the usual problem to which the Variate Difference Method is applied 
it is' assumed that after taking a sufficient number of differences we shall approach 
a state in which the corresponding values of Xt and Yt, the I'esiduals left after 
the ordinates of an ;ith order parabola have been subtracted from Xt and yt, are 
mutually at random in time or space ; or that 
X yPp = 0> xPp = 0, yPp = 0, 
for all values of p other than zero, and that 
xpn = 1 = rPo , X yP« = 'A-y > 
i.e. the con-elation between A^, and Yf Upon this assumption it follows at once 
from the modified form of (xvii) that 
nR = XYPii or 'A^^.r^ A„?/( ~ ''a'F' 
the fundamental relation of the original Variate Difference Correlation Method. 
Let us now turn to the particular type of problem in which we wish to corre- 
late the successive values of the same variate. If we are correlating the values at 
intervals of k, we shall have as corresponding variables, not Xt and yt but 
yt and yt+k so that 
xiiPj-n becomes pj+u-n and xvPj-n may be written pj+k-J"'' 
xPj—n )j Pj—n >i xPj—n d d Pj—n'"^ 
yPj-n „ Pj-n » YPj-n ,7 ,, Pj-n"^ 
where as in the notation of page 35 pp is the correlation of successive values of 
the variate at intervals of 2^> and /o^^'"' the correlation of successive residuals 
(at intervals of p) which are left after the subtraction of the ordinates of an 
nth order parabola representing the secular change. Hence we have from equation 
(xvii) that nRk . oi" the correlation between the ?ith forward differences of yt and yt^k 
is given by 
2 III 
1 (- lY+> /Ox-^;-.. 
