Egon S. Pearson 
41 
The results of the analysis given in the three illustrative problems below will 
be used in obtaining the values of various constants in the reduction of the 
experiments in the later sections. It seemed desirable to collect the algebra 
together in this way, but in reading this paper the reader may find it more 
convenient to pass on and refer back to the theory when occasion arises for the 
numerical application of the results. 
Problem 1. In this and the following illustrations of the method of the 
preceding section, the notation of Section IV fur the correlation of judgment will 
be used. 
I shall suppose that we have 7n series of observations through the course of 
which there is some form of secular change ; the means of the different series, or 
the values of pd, varying considerably. The coefficients of correlation for the 
combined series, R,, R.,, . . . . . . have been calculated, and also the single 
coefficient R/, the correlation of the successive values of the observations (at 
intervals of 1) after the series means have been fitted together — i.e. after removal 
of secular change. 
It is clear that AjT/^ = AjlV, where yt = + Y/, within any one series, and 
n n 
1 t (A,7/, . A,2/i+i) = S 2 (A, Y't . A, 1^',+,) etc., 
VI t=l III /--I 
where S again stands for summation for the m series, so that the 1st difference 
m 
correlation equations (xviii) and (xix) are applicable, and become 
P — 1 + 2Ri — Ro — Rk-i + 2Ri — Rjc+i / o + „ i / \ 
2(1-R,) ' 2(1-R,) /-2,to.-l ...(XX), 
1 + 2R,' — Ro' — R'jt— 1 + 2Rfc' — R 
k=2, to s - 1 ...(xxi). 
2(1-R;) 2(1-R/) 
From (xx) we get the values of ^R^, A' = 1, 2 ... s — 1, and using these and value 
of R/ already supposed to be known, the s — 1 equations (xxi) will give the s ~ 1 
unknowns R,', . . . R/. 
The accuracy of this method will of course depend on the errors involved in 
the assumptions (a), {b), and (c) of page 37 above. 
Problem 2. To obtain the coefficients of correlation of the successive residuals 
left after the ordinates of the " best " fitting straight lines have been subtracted 
from each of vi series of observations, that is, after the removal of a linear sessional 
change as well as a secular change. In the notation of p. 35 these coefficients 
may therefore be written 
Rj , R.2 ■ . . R/t — 
In the first place let us obtain the constants of the straight line " best " fitting 
the 50 observations of Group 1 of a series ; this can be done by the method of 
Least Squares. 
If for any series the equation to the line is 
y = d + b (^t— "-^'^ — ^0 before) (xxii). 
