250 Oil Correcfiojhs for Monienf -Coefficients 
We have for the " frequencies" : 
X 
Frequency 
0 to 1 
1-0U(),0000 
1 to 2 
•414,2136 
2 to 3 
•317,8372 
3 to 4 
•267,9492 
4 to 5 
•236,0680 
5 to 6 
■213,4217 
6 to 7 
•196,2616 
7 to 8 
•182,6758 
8 to 9 
■171,5729 
9 to 10 
•162,2777 
Total 3^162,2777 
Further, for small terminal 
subranges, we have : 
0 to ■a 
■2 to -A 
■4 to ■& 
•6 to -8 
■8 to •lO 
Frequency 
■447,2136 
■185,2419 
■142,1412 
•119,8305 
■105,5728 
It will be sufficient to take the subranges unity at the other terminal, or /<//;„ = 5 
h/lij)=l. Thus we have 
I// = 3-894,907, 
7',' = 20-016,109, 
"/ = 
•1414,2136, 
aj = - -2332,9561, 
= - 1-166,4781, 
11.2 = 
•0585,7863, 
ao = + -2583,1707, 
a.,' 
= -f 6-457,9263, 
11 :! = 
■0449,48!)9, 
- -2657,4026, 
a.; 
= - 33-217,5325, 
Ih = 
•0378,9373, 
«, = + ^1798,6052, 
= + 112-412,8250, 
n-J = 
•0333,8505, 
(,, = -•0586,1091, 
= -183-159,0938. 
Actual values 
"V- 
_j = •0674,8987, 
/;/ = 6i = + ^050,0105 
+ -0500,0000, 
ii'p- 
0620,6337, 
b.: = h, = + •002,4585 
+ -0025,0000, 
n'p- 
= -0577,6716, 
b,' = b., = + -(mAHi7 
+ -0003,7500, 
n'p^ 
_i = -0542,5612, 
h; = b, = - -000,0497 
+ -0000,9375, 
= -0513,1672, 
6/ = 6, = + -000,1316 
+ -0000,3281. 
These values of a's and b's lead to the abruptness functions : 
iL(a/ - + ^J^^a^) = - '057,1279, - ^i, {a.! - ^t^t^/) = - 
(W - 6^ V + ^6/) = -004,1669, (b.: - ^5^6/) = 
Accordingly we find 
^/ = 3-394,9066 - 052,9616 = 3^341,9450 
and ya,/ = 20016,1090 - 083,3333 + -066,7155 = 19-999,4912, 
-016,6425, 
-000,0205. 
which gives 
For comparison we have 
f^., = 8-830,8953. 
Raw moments 
3-3949 
20-0161 
8-4907 
2-9139 
Using only 
Sheppard's 
corrections 
3-3949 
19-9328 
8-4074 
2-8995 
Full corrections 
3-3419 
19-9995 
8-8309 
2-9718 
True values 
3-3333 
20-0000 
8-8889 
2-9814 
It will be seen that Sheppard's corrections alone are worse than the raw moment 
results. In other words they should certainly not be used alone for J-shaped curves; 
it would be better to take the raw moment results without any corrections. On the 
other hand the full corrections even in this extreme case — where (a) the Euler- 
