Miscellanea 
443 
The first summation on the right is zero ; hence 
f Run + l s t . of fifn + l \ *V I / x :_\ 
l-^B + l, h + 1 a t \ Il n + + 1/ °t ' 
This with a slight difference of notation is the result obtained on the Gaussian hypothesis*. 
The above proofs justify the statement that the general selection formulae given by me are 
independent of any Gaussian assumption. They are really peculiar to the general idea of the 
manifold linear variate u which gives the maximum correlation coefficient of an (?i + l)th 
variate with n other variates. They do not involve any idea of continuity or any hypothesis as 
to the nature of the selected means, standard deviations and correlations beyond the funda- 
mental assumption that the selected population really exists inside the unselected population. 
There need be no hesitation therefore in applying these formulae to any cases whatever in 
which the correlation coefficients have valid application at all. 
* Phil. Trans. Vol. 200, A, p. 17, Equation (xlvii). S in our present notation is a summation in (xix) 
of every value, but t, of t'. In the Phil. Trans, paper S x is a summation for all values of t': see p. 18. 
VIII. On a Fallacious Proof of Sheppard's Correction. 
The ordinary proofs of Sheppard's corrections for the moments are somewhat lengthy and 
depend entirely on the principle of high contact at the terminals. Mr G. U. Yule in his recent 
Theory of Statistics, p. 208, has given a proof in a few lines which is absolutely independent of 
this principle, and which from its very simplicity is likely, if not criticised, to be generally 
adopted. Unfortunately it is wholly fallacious. The error lies in the words "the correlation 
between X and 8 is zero, for the mean value of 8 is zero for every interval." What Mr Yule 
should have said is that the correlation between his Z and 8 is zero, and he should have reached 
the conclusion 
tri 2 =<J 2 + $iC\ 
and not ""i 2 = <r 2 - T V c 2 , 
for he is really working out the mean square for the histogram and not the true figure. He 
would thus have failed to obtain the correct value, which he does not appear to recognise arises 
solely from the fact that the ' trapezettes ' cannot be treated as rectangles. In the case of curves 
of frequency without terminal contact, Sheppard's corrections are not the proper ones, and their 
general adoption without regard to their limitations is to be deprecated. Such adoption is 
directly encouraged by a fallacious proof of the above character. K. P. 
56—2 
