94 On the Variations in Personal Equation 
and it follows that the relation for Ri' can be exjjressed in the form 
= Pk + h Ra", 
which must be compared with the relation 
Ryt' = p + q . (xlviii) bis, 
where ' p = -1524<, = '6817, /• = -7l05, 
that has been found empirically t(j fit the actual values of Rjt'. 
If the expressions j)^. and 4 were constant for 1, 2 ... 14 an interpretation 
of (Hi) would be at once suggested. Namely that R^", the coefficient of correlation 
of the successive residuals Yf and Yf^^ It'i't after the removal of the- secular and 
sessional changes is expressible in the form 
= (liii), 
that is to say, making allowance for the presence of accidental errors, the law of 
relationship between the successive estimates suggested on p. 90 above, holds 
good. Now without finding the curve which represents the sessional change in 
each series we do not know the values of S^" and 0^. We have however that 
S-^ + GiJ^^S^' (liv), 
where Sk is the standard deviation of the observations in the kth groups after the 
removal of the secular term. The values of Sk are given in Table V, p. 58 ; 
they are seen to increase as k increases and therefore pk and can only be 
constant for all values of /.■ if 
;SV'" *SV'- >S'i4' - , 
TTT =7Ti; = ••• =7VT Ov- 
Lti (jr.) U"i4 
That the relations (Iv) should hold approximately is not at all improbable; for 
with a sessional change of the parabolic form of the curve (xxx) illustrated in 
Figure 4, the standard deviations of the ordinates in the later groups will increase 
owing to the increasing drop of the curve towards the end of the series while Sk" 
may increase with k owing to greater variation towards the end of a session. 
In fact for this particular mean series with its sessional curve represented by 
(xxx) it is found that 
G, = -0336 ins., G,, = "0406 ins., 
while >S/' = -0165 ins., ,V = -0201 ins., 
that is to say, the variations superimposed upon the main sessional change (the 
distances of the points plotted in Figure 4 from the parabola) become greater 
towards the end of the series when the observer's judgment perhaps became more 
S " S " 
erratic as he grew tired. These values give ~ = '49, -~ = -50 suggesting that 
the relations (Iv) do hold very closely. What we find therefore in this typical mean 
