276 
PEOFESSOE K. PEAESON ON THE MATHEMATICAL THEOEY 
Accordingly is < S/Xj" by a considerable amount, and the observations are 
immensely flatter than the normal curve with which they can be fltted. Actually 
the normal curve has a maximum ordinate which rises some 15 to 20 per cent, above 
the corresponding ordinate of the observations. Hence quite apart from the question 
of equal negative and positive errors, we should assert that because /x^ = S/u,/ is not 
sensibly satisfled, it follows that one or other or both the axioms (/3), (y) cannot be 
true for this distrilmtion of hits. 
I propose to look a little more closely into the probable errors of the quantities 
connected with a normal distribution. I take d to be the distance from the mean to 
tlie mode, and deflne the skewness, Sk., as in earlier memoirs, to be the ratio of d 
to cr. I write Mr. Filon and I have dealt with the 
probable errors of skew frequency curves in a special memoir,* and deduced those for 
the normal curve as the limit to those of a skew curve of what I have termed 
Type III. The disadvantage of this procedure is that it supposes the deviations 
from symmetry to take place along a class of curve for which 
2P3('W — /^4) + 3/X3" = 0.(xix.). 
is thus known in terms of and /Xg. The result is that when /ixg is put zero to 
reach the normal case, the error of /x.^ is found to be absolutely correlated with that 
of /Xo, and the probable value of this error to be deducible from that of by means 
of the relation 
= 3/X2^. 
To obtain perfectly general results we must use not Type III., but Type I. or 
Type IV. of that memoir, curves in which /Xg, p-g, and /X 4 are absolutely independent 
of each other. Our results can easily be deduced by aid of certain elegant formulfe 
due to Mr. W. F. Sheppard.! In our notation these are :— 
Probable error of /x^ = '67449 is given by 
/xy 4- P" fj?f~\ /Xg 
(xx.). 
= Correlation of errors in /x^ and /x^ is given by 
p _ H-p+j P^^p-i /^y-fi 7/^jo+i ~t p(lfip-\ f^q -1 /Xq /XpR? /wi 1 
These results are perfectly general whatever be the law of the frequency. As 
special cases we have, when = 0 : 
* ‘Phil. Trans.,’ A, vol. 191, pp. 229-311, especially p. 276. 
t ‘Phil. Trans.,’A, vol. 192, p. 126. 
