EctOn S. Pearson 93 
Then if 2 indicates summation for the m (or 20) series, n = 50, the number 
m 
of observations in each group of a series, and k takes any of the group numbers 
>i 
1, 2, ... 14, since y = fp (t) will be the " best " fitting curve of its type 2 Yt+k-i = 0 
approximately, and on combining the ??i series 
2 2 (F,+,_o = o. 
m f=l 
Again we have no reason to suppose that there will be any correlation between 
the sessional term fp{t) and the residual Yt, so that 
2 2 {F,+,_,/;.(< + ^''- 1)1 = 0' 
for all values of ^' and k' between 1 and 14. 
As i/t' is freed from the secular term, using the relations above we have that 
2 I {(fAt)+Y,)(f,(t + k)+ IV.)} - -"S 2 {4i^l2 2 
m t = l III f = l I I TO t = l { 
Jh 2 {Mt)+Y,f-mnhi'^^\ W i{Mt+k)+Y,,,r-mnk 2 ^^1^ 
V \_mt = l {mt = \ mn ) J ; = i [m t = l mn 
(li), 
Rfc Si 'Sjc+i ' + Fk 
where R^" is the coefficient of correlation between Yt and Yt+k, S^" and Sk+i" are 
the standard deviations of the F's of Groups 1 and A; + 1 (see (xi) and (xii) on 
page 35), and 
7^, ^ -L 2 2 m, it + k) - |2 2 |2 2 ^^^'f 
(?.^ = -i^2 2 {/,(i + ^-l)}^-{2 2^'^±Alll)r 
It will be seen that Gk is the standard deviation of the ordinates of the curves 
representing the sessional changes, y = /p {t), which correspond to the observations 
F]c . 
in the kth. groups, while — is the correlation of these successive ordinates at 
(Ti (jTk+X 
intervals of k. If the sessional changes were linear this correlation would be unity, 
and a little consideration will show that if the sessional change in each series can 
be represented by a curve of gradual bend, the correlation will not be fer from 
this value. For example in the case of the parabola (Equation (xxx), p. 47) 
which was fitted to the mean sessional change and is drawn in Figure 4, it is 
found that 
n ri ^ "994. 
We shall therefore make no great error in assuming that 
Fk = G^i , 
