EgON S. Pp]ARSON 
59 
(h) The Combination of Series. 
Having discussed the reduction of the individual series, I will proceed to con- 
sider the results of combining the 20 series. The formulae (v) to (viii) on pp. 33 and 
34 give the values of Bk, Sk, and R^. which are tabled below. Remembering that 
A and /Sf] are the mean and standard deviation of the combined observations 
yi, 2/2 ••• 2/50 from each of the 20 series, D., and the mean and standard deviation 
of the combined observations 1/2, ?h ■■■ 'l/r,i, and finally Du and *S'i,i of ij^, y^,, ... 7/,,,, we 
see that the progressive decrease in Dk as k increases indicates the shortening in 
the estimate of a third during the course of a sitting, while the increase in Sk may 
perhaps be partly due to increasing variability of judgment due to fiitigue. The 
values of R^- are large, but this is to be expected owing to the large changes in 
personal equation from series to series; in foct for k = 13 it will be found tliat the 
limiting expression Lk of page 34 gives 
Z,, = + -5435, while Rj.^-Glol. 
The reason for this difference between Ly. and R^; is that 2 {pr.:<^i(ru), and therefore 
m 
R13', does not vanish. The next step is t(j obtain the values of Sk and R^,.', or the 
standard deviations and correlations of successive judgments after the secular 
change has been removed. They are found from Equations (ix) and (x) of p. 34 
and are given in Table V (5th and Gth rows). 
There is here an opportunity of testing the accuracy of the Difference Correla- 
tion method discussed in Section V (b) ; the case is that of Problem 1, page 41, the 
values of Ri, Rj-.-Rnj are known and give the correlation of 1st differences, 
jfii, ii?2 ••• 1-^12; these together with the coefficients of correlation of 2nd differences 
to be used later, are given at the bottom of Table V. Tlaen using the value + '6246 
for R/, we get the values of the 12 quantities Ro'... Hy/, which have been inserted 
in the 7th row of Table V. It will be seen that the values obtained by this 
approximate method agree well with the others, the differences being within the 
probable error of the R/'s up to and including R;,'; beyond this the approximate 
values become rapidly too small, the error, from the form of Equations (xx) and 
(xxi), being clearly cumulative. This failure is certainly largely due to the fact that 
the errors involved in the assumptions (a), {b) and (c) of p. 37 are not negligible 
when the later groups enter into the correlation, for we have already seen that 
both Dk and Sk change steadily with k. 
The values of Sk and R^/ in Tiible V correspond to the average values of the 
standard deviations and correlations of successive judgments in the individual 
series, i.e. of the a's and p's given in Tables III and I. Owing to the sessional 
change which occurs during the course of nearly all the series, R^' does not vanish 
as k increases, but appears to approach a limiting value in the neighbourhood of 
+ "16. By obtaining for the separate series, the coefBcient of correlation, p/, of the 
successive observations at intervals of one, freed from the linear sessional change, 
a step has been made towards t"he further reduction of the problem. R/', the 
