26 
On the Variations in Personal Equation 
for any one session, and <^ (t^) may be described as the secular term in the ob- 
servations of the jijth session. It remains therefor'e to consider the function /p (f). 
Supposing that there were n observations made in a session, it would of 
coui se be possible to fit an (?; — l)th order parabola on which all the observations 
would lie, so that the values of Yf would all be zero, but such a curve would be 
entirely useless. If the observations are made at finite intervals so that we can 
imagine that one may be interpolated between two others, owing to the mass of 
random errors to which each judgment is subject, we should not for a moment 
expect that the interpolated error would lie on, or even close to the (?i — l)th 
order parabola. A curve of far lower order would probably give a much better fit. 
If the sessional change is a sign of some physiological change of state which is 
affecting the observer's judgment, it is natural to suppose that it can be repre- 
sented fairly ch^sely by some simple curve — a low order parabola if not a straight 
line, or perhaps, if periodic, a sine curve. Suppose that in a practical case, a 
first or second order parabola has been fitted to the observations of a session ; then 
it will be easy to test whether the residuals Yt follow a Gaussian distribution; 
a simple practically sufficient, if not theoretically sufficient test would be to find 
whether 
2 (7t) = 0, S (F,«) = 0 (ii) 
M4 5;(>V) 
approximately. 
But there is a further possibility ; it may be found that although the relations 
(ii) and (iii) hold approximately, the F,'s are not randomly distributed in time, 
and that there is in fact a correlation between the successive values of Yt, so that 
S(F,1V,) 
't Hi- 
^'l'^ - /«"-^ V - ^ ^ 
for perhaps several positive integral values of h from 1 upwards. 
To emphasise the importance of the different terms in the relation 
vVt = 4> (Tp) +fp{t) + Yi (i) bis, 
let us take the case of an astronomer who makes a number of observations, often 
at man}^ days' interval. He will take a mean 
T/ = mean (f> (t^,) + mean f^j (t), 
but he must not suppose that the quantities 
pVt -y = 4> i'Tp) - mean ^ (t^) + /^, (t) - mean/^, (t) + F, 
follow a Gaussian distribution. It will be only a part of the expression that does 
so, the Ff's, and it is possible that even these may not be true. 
Further it is clear that successive values of — y will not be independent ; 
correlation will arise from the inclusion of both the secular and sessional terms, 
