Eleanor Pairman and Karl Pearson 
241 
We divide the space from a; = 0 to a-' = 10 into ten subranges giving the following 
system of " frequencies " : 
Absolute 
frequencies 
66,667 
121,895 
157,848 
186,923 
212,023 
234,440 
254,888 
273,811 
291,505 
308,185 
Total frequency = 2,108,185 
np_i 
Proportional 
frequencies 
031,623 
057,820 
074,874 
088,665 
100,571 
111,205 
120,904 
129,880 
138,273 
n'p-3 
n'p-i = 
n',; =-146,185 
1-000,000 
These lead to 
Abruptness Differentials and Ordinate 
Calculated 
Actual 
«! = 
-•0131,0643 
?/„ = 27630-78 
0 
«2 = 
-•0444,8167 
V,,' = +93775-59 
oo 
03 = 
•0258,4150 
V„" = - 54478-66 
QO 
a4 = 
--0148,8400 
V„"'= +31378-23 
00 
«5 = 
-0045,0200 
■?/„iv=_ 9491-05 
00 
Now it is clear that the ordinate of our auxiliary curve is not zero, but it looks 
larger than it really is relative to the ordinate at the other terminal which is 
316227-77 so that the ratio is only '087, or if the curve be actually drawn to any 
reasonable scale, the ordinate of the auxiliary curve at the vertex which is less than 
one-tenth of that at the other terminal, looks relatively small. We may also com- 
pare it with the ordinate of the actual curve at a; = 1 which is 100,000, or between 
three and four times as great. Similarly the abruptness differentials are not infinite, 
but their values in the actual curve are very considerable at a; = 1 and are then : 
2// = 50,000, 3/1" = - 25,000, 2//" = 37,500 and y/^ = - 93,750. Thus the first two 
for the auxiliary curve are about double, the third of the same order, but the fourth 
is much less. Clearly all this is a result of the fact that we cannot expand Va; in 
a series of integer powers of x, and this is one of the reasons why we selected it. 
We want to determine whether the formulae give a very bad result for the moments 
even in the case of extreme abruptness. Accordingly our real test lies in the values 
of the deduced moment-coefficients and not in those of the abruptness differentials. 
We pass now to the 6's and find : 
61 = 
63= 
•1499,9857 
- •0074,9283 
- -0003,9450 
- -0000,2600 
- -0000,3800 
2/p 
Vp 
Vp 
Calculated 
= 316,224^74 
= - 15,796^27 
= - 83r68 
'= - 54-81 
80-11 
Actual 
316,227-77 
- 15,811-39 
790-57 
118-59 
29-65 
Here again the values of the terminal ordinate and the abruptness coefficients, 
although good in the former case, are only approximations and the real test must 
