FOUNDATIONS OF THEORETICAL STATISTICS. 
319 
Inserting the values 1, 2, 3 and 4 for k, we obtain for the aperiodic terms of the four 
moments of the grouped population 
r* 
iA 0 = | xf (x) dx, 
J — 00 
2A ° = 1-00 ( X ' + ^ dX ’ 
A — I 
J — C 
4^-u == 
x* + dx, 
x ^ + f\ f(x)dx . 
If we ignore the periodic terms, these equations lead to the ordinary Sheppard 
corrections for the second and fourth moment. The nature of the approximation involved 
is brought out by the periodic terms. In the absence of high contact at the ends of the 
curve, the contribution of these will, of course, include the terms given in a recent paper 
by Pearson (8) ; but even with high contact it is of interest to see for what degree of 
coarseness of grouping the periodic terms become sensible. 
Now 
i p = * r 
A s = - Z \ 
7T__ Jo Jt- 5 “ 
sin d£ | £ k f{x) dx, 
2 7rs£ 
2n ' rt+ia 
sin s9 d6 £ k f (x) dx, 
But 
therefore 
2 
a J-c 
2 
a 
2 7 rSg rUi(l 
a " Jf-ia 
rx+ia 
f (x) dx g k sin d£. 
oo J x—ia CL 
2 * s tdi = 
,A S =(-)'■“— f cos — -f(x)dx; 
7T.S* J - oo CL 
2 (*+*» ^ . 
£ sm 
CL Jx-ia a 
a 27 rsx 
— COS-COS 7 rS, 
7 rS CL 
similarly the other terms of the different moments may be calculated. 
For a normal curve referred to the true mean 
Q s 2 <r 2 
,A S = (-)•« f <T-, 
iB s = 0 , 
in"which 
CL = 'lire. 
The error of the mean is therefore 
/ cr~ 4 <r 2 9cr 2 \ 
— 2e( e 2(2 sin 0—\e 2 ‘ 2 sin 20 + ^e 2 * 2 sin 3 0—... J. 
2 Y 2 
