Miscella7iea 
309 
Now it is a well-known theorem due to Euler and Maclaurin* that if /(^.■) be any function 
of X, 
j/(.r) dx = ^hf{x,) - /(.^■)+ §/' (■'•) - Iyq^'" + 
whei'e the term in brackets is to be given its values at the limits of the integration on the left. 
Hence if /(.v)) /'(■'')> /'"(■*) etc., all vanish at the ends of the range of values under consideration, 
we have simply 
2/*/ (.?■,) = ^f{.'€)dx (iv). 
Now suppose are functions which for different values of p and .s are such that they 
and their differential coefficients vanish at the ends of the range of frequency. Then 
2 {«,. J <t> (.r) x'dx+^^ j 4>" (.r) ■^''d.r + ^^^^ j 4>^^ (..■) dx+ (v). 
But by integrating by parts 
j r (*•) -v'dx = [0 '> - 1 {X) x^ - ,.0 r. - 2 - 1 + , (, _ 1 ) 0 P - ^ (.,) 2 + . . .] ( vi ). 
This clearly vanishes at both limits if be greater than s. Hence no term of (v) need be 
retained for which p is greater than s. 
Put s successively 0, 1, 2, 3, 4, we find 
2(«,) ^jcj,{x)dx^F, 
2 (Mr-r,.) = j<t> (*') '^'d-V = ^Mi', 
2 {n,x/) = j 0 (,r) x^-dx + 2 ^J^=N (^.;+ h'^ , 
2 {n,x,?) =jct> (..■) xMx + G^j<t>{x) xd.v = # (^f.,' + ^^t) , 
2 («,„r/) ^jcjy (..) x*dx+^-^ j 0 {X) x^dx + N= N (m/ + \ /'■>,' + ^ /'^) ( vii). 
Here /i/, /^j', fij', /^j' multiplied by N are the true first four moments al>out the axis of y of the 
frequency curve. If we take the axis of y through the mean, we find /ii' = 0, and if we write for 
the moments about the mean iVjug, iV^j, N|x^ and Nv,^,, Nv.^, Nv^ for the moments of the 
frequencies about the mean, we find, since 2 (»,..??,.') = iVi^^, 
Mo = fu = l>^ /ii = i^i = 0 I 
''2 = "2- ^3 = ^3 [ (viii). 
A2 7A< 
= 2 "2 + 240 
In other words the area, the centroid and the third moment are not changed by using group 
frequencies for ordinates, and from the second and fourth moments we have to subtract 
r^'' and ^AS-^A^ 
respectively, supposing the raw moments already referred to the mean. These are the well- 
known Sheppard's corrections. 
* Boole's Finite Differences, Chapter V, The expansion of course depends on /(.r) satisfying 
continuity conditions. 
