MISCELLANEA. 
On an Elementary Proof of Sheppard's Formulae for correcting 
Raw Moments and on other allied Points. 
[EDITORIAL.] 
Several biometricians having expressed difficulties to the editors concerning the proof 
and use of Sheppard's corrections in calculating moments, we venture to publish the following 
elementary consideration of the subject taken from manuscript notes of the past few years. 
For the complete treatment of the subject the reader must always refer to the original paper*. 
Let the equation to the frequency distribution be 
ybx being the frequency between x and x + bx. Let h be the base unit for grouping the raw 
material. Let N be the total frequency and n,. the frequency on the rth base unit h, between 
x^. — \h and Let y,! be any ordinate corresponding to x^ + x'. Then 
Now if (fiix) be a continuous function which can be expanded by Taylor's Theorem, i.e. 
does not become infinite within the range used : 
{"^ ^-'''-^ + ^■^'•^ +ri ^-''-^ + •••} ^' 
= (-^v) + ^ 4>" i-vr) + ^9^0 "i"" (■^■'•) + (i)- 
Now X,. is clearly here any abscissa of the frequency curve and this expression gives the 
frequency on a strip taken anywhere, provided it has a base h and mid-abscissa x,.. It follows 
that if 2 denote a sum for all values of r, 
2 (»,. (.^■,)■') = A2 (c/> (x,) {x,y) + 1 2 {cf>" (x,) {x,)') + 2 (<^i^' (x,) (.r,)") + etc (ii), 
where .9 is any positive integer. 
W. F. Sheppard : " On the Calculation of the most Probable Values of Frequency Constants, 
for Data arranged according to Equidistant Divisions of a Scale." Proc. Loiul. Math. Soc. Vol. xxix, 
p. 3-53 et seq. 
