OF ERRORS OF JUDGMENT AND ON THE PERSONAL EQUATION. 
279 
The result that the probable error of is zero, but that of is finite, may 
appear paradoxical, but ir is due to the fact that the errors are treated as small 
quantities and Involves /u. 3 ^, the square of a small quantity, or one zero, if the 
distribution were truly normal. 
Of these results, Mr. Filon and I have already published with a less general proof 
(xxiii.), (xxiv.), (xxvi.), (xxvii.), (xlii.), and (xliii.). Instead of (xxv.) and (xxviii.), 
we found 
Probable error of = ‘67449 and = 1 , 
i.e., replacing ^96 = 9‘8 by y/ 72 = 8‘5, and ‘866 by 1 . This was due to the fact 
that we considered the variations from normality to be given by a distribution of 
Type III. (see p. 276). The differences, however, are not such as to invalidate argu¬ 
ments based on the general order of the probable error of 
We have now general relations enough to answer the following questions 
(i.) Does the value of d found from (xxx.) differ from zero by an amount large as 
compared with the probable error of d given in (xlii.) ? 
(ii.) Does the skewness found from (xxxi.) differ from zero by an amount large as 
compared with the probable error of the skewness as given in (xliii.) ? 
(iii.) Does the value of /ig differ from zero by a value large as compared with the 
probable error of yug given in (xxiv.) ? 
In all these questions we have a test of whether the distribution is really a random 
selection from a symmetrical distribution, i,e., from material obeying axiom (a). The 
same thing is again dealt with by testing the error of as given by (xL). 
(iv.) Is the condition fx^=- 3/Xo^ or = satisfied for the distribution, i.e., does 
/S 3 differ from 3 by a quantity which is not large as compared with its probable error 
as given by (xxxix.) ? 
If (i.) to (iii.) are satisfied, but not (iv.), the distribution is still not a random selection 
from material obeying the normal law, i.e., axioms (/3) and (y) cannot both be true 
for it. 
Lastly, if the material does not obey the normal law, does the criterion K differ 
sensibly from zero, and therefore form a characteristic to be regarded ? 
I turn first to the motion of the bright line and give in Table XVII. the constants 
for this series of distributions. 
See ‘Phil. Trans.,’ A, vol. 191, p. 277. 
