90 
On the Variations in Personal Equation 
nor with the fundamental part of the judgment ««, we shall have the following 
approximate relations 
where N is large compared ^ 
with k 
1 
1 
N 
-0 for A: = 1,2, 3,..., 
0 
} (xlvi). 
where k and k' take any of the 
values 1, 2. 3,... etc. 
But the correlation between successive values of the ys at intervals of k is 
Pk- 
N 
\/|i<«' ^ " * (I, --/')] i,l<"«- + - * (I, 
N 
< = 1 / = l -i* /-l -tV 
in view of the relations (xlvi) 
where [o(;a(+i] is the first order product moment coefficient referred to mean of the 
successive as at intervals of k, 
and \/a/;- is the standard deviation of a,, 0,+,, ... ak+x, 
^^k' „ >. " /3k, /3k+i, ■■■ ^k+N, 
'^"•^ J oik' + (3k- >. Vk , ?/k+i , ■ ■ • yk+N- 
Now unless there is a steady sessional change in the as, we may assume that 
for large values of N 
= a..- = ... = OLk" = ... = a.-, say, 
and similarly unless the accidental errors are steadily increasing or decreasing in 
magnitude _ _ _ 
n 1 [c(taf+k] a' [o(tO(t+k] 
and we have = = ■ — = _ •^a,,a,.r.- 
But on the assumption made above of zero partial correlation between two 
estimates which are not consecutive, we have found that ?aj, ai^,, the correlation 
between the observer's real estimates at intervals of A;, can be expressed in the 
form r*^, and therefore 
a^ + /3' 
.(xlvii), 
