42 On the Variatio7is in Personal Equation 
where the tth. observation is 
n 
K= S I? 
^ = 0 and — = 0, 
da do 
we have that 
therefore 
to be a minimum, 
or 
and 
S Yt = 0 whence S = iid, 
,aH-H-"r)}V"r 
= 0 
2 yt t 
n + 1 
1=1 
Or, the first order product moment coefficient about the mean of yi and t 
Pn 
= 6 
12 
giving for the constants of the best fitting line 
1 " 
n t=i 
The next step is to obtain the correlation of the successive residuals left after 
the ordinates of this line have been subtracted from the observations. 
We shall have that 
?i + 1^ 
2 
- nd (d + 
= d + yt+,) - lid- + h 2^ + 1 :^ 1 j Ojt+i - d) 
^){yt-d) 
n + 1 
¥S it- 
t = i 
n + l 
n-1 
+ t Yt Y,+, - luP - d {yn+, - 2/0 
n + l 
11- 1 
= S Ff Yt+, + b ]2)ipn + — (y„+i -d) + {y, - d) - y„+, + y. 
2 \tr-iit + ^- 
!" = ! ( 4 
» ( 91 + 1 ?l — 1 
= 2 Ft F,+, + h j 2Hj(;„ - nd + - 2/i + — 2~ 2/n+i 
^=1 
•6^ 
12 
6= 
^ 2 YtYt+, + .~-.n{n'^-l) + h , g 
