OF EREORS OF JUDGMENT AND ON THE PERSONAL EQUATION. 
241 
personal equations from his knowledge of the relative personal equations. How then is 
he to measure the relative goodness of observers ? In turning this problem over in my 
mind it occurred to me that if there were no means of measuring the average absolute 
error of an observer short of an experiment ad hoc* still, if we could deal with three 
observers, their relative variabilities would give us the means of determining their 
absolute variabilities, and the astronomer or physicist would thus really be in a 
position to judge something about the steadiness in absolute judgment of a series of 
observers. He could find their o-q^, ctqo, and 0 -^, 3 , and so determine their stabilities. 
Now, if one accejots the independence of the judgments of independent observers, 
(vii.) follow at once, and we have an important problem simply solved. I therefore 
organised a series of experiments to illustrate (vii.), but instead of discovering a new 
method of testing observers’ stability of judgment, I found that (vi.) did not hold ; 
that, indeed, could be smaller than both o-qi and ctqo, or, in other words, that the 
judgments of independent observers could be sensibly correlated ! I accordingly felt 
compelled to discard the current theory entirely, and develop one in which the 
correlations like Cjo, &c., are not supposed to be zero. Before describing this, 
however, I must point out that even If, on the ordinary view, we put these correlations 
zero, we ought to expect correlation in the judgments of observers when they are 
referred to the judgment of a standard observer. 
This may be proved thus :—- 
Let 7)^ = Pqy — [X] — f), with similar values for and 173 . Then S ( 77 ^) = S (> 7 .,) 
= ^ (^ 3 ) = 0 ; S ( 773 “) = ; 8 (770 773 ) = n<j(y 2 o-Qg = 0 , since = 0 , and similarly 
8 ( 77377 ,) and 8 ( 77 ^ 770 ) = 0. 
From (iii.) we have : 
_ {(p%\ ~t jt)I j^03 ~t ^3 V\) ( itu ~t j^fl2 ~t ^3 '^ 2 )} _ ^ 
Pqj ]0 - - j 
remembering that pz\ = Bos ~ ibn B 33 — Bos — Bon relations cited above. 
Hence : 
Similarly :t 
Pzi 12 — <^03V(^31^32) J 
d2) 31 — ^03V(‘^21^23) > ^ 
do 23 — ^ofiA^li^ls) 
. (viiL). 
These expressions can never vanish, and tlius, if the current theory were true, the 
judgments of two observers referred to a third as standard would undoubtedly be 
* As, for example, by an artificial star, whose actual position at each instant of time is known, first, I 
think, used by N. C. WoLFF in 1865. Unfortunately the personal equation seems to vary a good deal 
with the speed and intensity of the star observed. 
t Of course relations of the type a-is^ = cros^ + o-yy will also hold by (vi.) if there be no correlation of 
absolute judgments. 
VOL, CXCVTII.—A, 2 I 
