242 
On Corrections for Moment-Coefficieuts 
depend on the moment-coefficients. If we go as far as ^u/ we have to calcialate 
the following eight expressions : 
- 6^"/ + 25Vo«5' = - -0131,0643 - -0004,3069 + -0000,017!) = - -0135,3533, 
- Tfir"/ = - -0444,8167 + -0005,9063 = - -0438,9104, 
a/ - ^aJ + ^i^a; = - -0131,0643 - -0020,5092 + '0000,1876 - - -0151,3859, 
a: - ^^a^ = - -0444,8167 + -0013,0235 = - -0431,7932. 
The quantities we require are 
tV ("/ - c/ii"'/ + ^Ao«/) = - -0011,2794, (a,' - ^^a/) = - -0003,6576, 
iV ("/ - o^a/ + ^i7T«/) = - -0003,7846, (^u - ^^«/) = - -0003,4269. 
It will be seen from these results that al does not contribute very much and 
a/ still less to the final corrections. We now take the ft's and find 
(6/ - J^ft; + W^o^'r,') = -1500,0513, {h: - ^t^V) = - -0074,9273, 
(^>/ - (/"V^/ + uio^/) =-1500,2972, {h.:--i^hi) =--0074,9055. 
Whence we deduce for the abruptness functions, since jj = 10 : 
-h (W - + ^-^j^W) = -0125,0043, Jj; (W - ^^6/ + ^^j,W) = -2500,0860, 
if' iW - + ^^uh:!) = 3-7501,2825, {W - J^t; + = 50-0017,1000, 
jio (^'Z - TYii^') = - -0000,6244, J^^; - j^yh:) = - -0001,8731, 
ioP' iW - jhW) = - -0374,6365, (W - ,4,6/ + ^ij^W) = "0037,5074, 
tVP - jkW + = -1500,2972, ^^iW - J^W) = - -0000,5945. 
We now give the values of the grouped naoment-coefticients about the origin. 
Alongside them we place their values as corrected by Sheppard's terms. We then 
give the values as found by full correction formulae and lastly the actual values as 
deduced bv intesfrating- the parabola. 
Values with full 
corrections Actual values 
.5-9994 6-0000 
42-8570 42-8.571 
333-3349 3333333 
2727-2757 2727-2729 
It will be seen that the fully corrected results are in most excellent accord with 
the actual values. Sheppard's corrections, although component parts of the general 
corrections, move if taken alone in the wrong direction, i.e. they lower moments, all 
of which need to be raised. Thus while Sheppard's correction lowers the fourth 
moment by about 21, our new corrections raise it by about 50, the result being the 
requisite raising by 29. 
It seems to us unlikely that a more unfavourable case for our abruptness 
coefficients could be found. It certainly euq^hasises the point that to obtain very 
Raw moments 
5-9880 vi 
42-6900 v.,'-^\, 
331 -0854 v-i - |i/V 
2698-7735 I'i'-h^I + ^u 
Values with 
Sheppard's 
corrections 
5-9880 
42-6067 
329-5884 
2677-4576 
* The a"s and the //'s'will iu this case be equal to the as and b's 
