Egon S. Pearson 
45 
the cumulative effect of this error may be considerable in the value found for 
R^" (s = 12, say). If then we take second differences 
1 
_ Iv yi - y-2 - yn+i + y„+o ] f^. yk+i - yk+^i - yk+n+i + yk+n+2 ] 
TO''- TO ^=1 
]^ fv -^1 ~ -^'^s ~ F„,_|_i + F„_|_2 l 1^ F^fc-i-i — ~ Ffc-)-;t4-i 4- Ffc^,i4.3 | 
= (Rk-," - 4R,-/' + 6R/ - 4R,+/' + Rit^/') 'S"^ 
and is independent of the differing values of the b's. 
The appropriate equations are in fact of type, 
p R/t-2" — 4Rfc_i" + 6Rfc" — 4Ri.+/' + Rfc+a" z^.-; ^\ 
oitt = ;r ; 77 77T (XXIX), 
' " 2(3-4R/'+Ro") ^ ^ 
for A; = 1, 2, 3 ... s — 2, where R_i" = R/' etc. and R„" = 1. Then using the known 
value of Ri", and that of R/', found as in Problem 2 from the first difference 
equation, these s — 2 equations will give the s — 2 unknowns R3" ... R/'. 
It is clear that similar methods could be applied in the case of sessional changes 
of higher order, but I have taken the algebra in these three Problems, as the 
results will be used in the reduction of the experiments later on. The general 
explanation and equations may have appeared long, but the actual calculation in 
any particular case of such quantities as iRi, i-fio, ... iRh, or oRi, ... iR/c, and then of 
R2', ...Rj', and R/', ...R^.", is exceedingly simple, and far shorter than a direct 
calculation from the crude figures would be. In two cases the correlations were 
calculated both by the difference correlation method and directly without approxi- 
mation, and the agreement of the former results with the latter established con- 
fidence in this method of approximation. 
VI. Experiment A (Trlsection). Reduction of Observations. 
(a) The individual Series. 
The observations of this Experiment have been reduced in more detail than in 
the other cases; the values of p^, k — 1, 2,... 13, were found separately for each 
series, and these and the values of d and a — the means and standard deviations 
of the Groups — are given in Tables I, II and III. Several points of interest will be 
noted; in the first place the observations have a marked tendency to decrease (i.e. 
for the estimate of a third to become smaller) both in the course of a series (as is 
seen by the general decrease of dh as k increases) and also in passing from the 
earlier to the later series. These are examples of what have been termed Sessional 
and Secular Changes. These changes are illustrated in Figure 3 where the centres 
of the circles give the values of for each Series, the length of the dotted lines 
from either side of these points representing the standard deviations o-j, and the 
