Miscellanea 
381 
If 1 = 2 this becomes 7(2 + 12 and if < = 4, A^ + ^^^+i^. We have however already noticed 
r M - i 
that if there is high contact the vakie of I {X — xY y^dx is given by the sum of the 
ordinates, i.e. the second moment is given by a series of which the general term is h'^y^^ and 
the fourth by one whose general term is ; hence if /x be written for the true and v for 
the unadjusted moment we have 
1 1 
and ^4 + -/x2 + ^ = i'4, 
1 
or V2- Y2""'^2, 
and ■'^-^''2 + 2lO = ''^- 
The odd moments are seen to require no modification. 
It is interesting to note that Formula II. will give the adjustment for the second moment 
11 3 
hut that for the fourth it gives i/i- - Tr: = lJ-A which only differs from the true result by - — - . 
" 2 " 24 240 
To demonstrate Sheppard's correction for the ?ith moment a parabola of at least the ;ith order 
must be used. 
There is another method of a very simple character by which adjustments when areas 
are given can be reached ; it consists of finding ordinates from the given areas and then working 
ou the values obtained. 
It is easy to obtain formulae suitable for our piu-pose in the following way : 
Let y;c = a + bx + cx^ + dx^ + ex"^, then 
2c 2e 
y,dx = A, =a + 24 + yg-o, 
, , , 26 80 , 242 
/: 
/; 
f - , , „, , 98 544 , 2882 
^ _^y^dx = A., = a-2b + ^^c-^^d + ^e. 
These equations can easily be solved and we obtain the following result : 
l4o {2134.'lo- 1 16 (.1 _i + .4 ^ i) +9 ^2)} XL, 
^=ii( -1360(^_i-^^i) + 200(4_.3-^+2)}, 
'h 26 80 242 
^j^dx = A_,=a-h 
^ A , 07. , 98 , 544 - , 2882 
iudx = A.i =a + 2o + — d-\ e. 
c = 
1920 
1 
1920 
{ - 2640 ^0 + 1440 (.1 _ , + .4 + 0 - 1 20 (4 _ 2 + ^1 + ■,)], 
'^^T^' 320(4_i-^^i)-160(^_2--4+2)}, 
^ = lio " - 1 + + 1) + -2 + ^ + 
