K. Pearson 
279 
It may be noticed also that* 
^(7= -693,580,83, or A = + -000,433,65, 
-692,930,49, or A = - 000,216,69. 
The latter is less divergent from the true value than the former, but they 
differ by as much as 1 in 3200 and 1 in 1600 respectively from the true value. On 
the other hand the worst of the above quadrature formulag (k) and (?/) give results 
only about 1 in 48,000 in error, while the best, like Boole's or Weddle's Rules, or 
(t) and (/J,), vary from about 1 in 6,000,000 to 1 in 17,000,000, while (^) and (tt) 
are almost as good. When we are dealing with frequency we probably never, and 
often when we are dealing with measurements, physical or economic, we do not, 
know our data with anything like the accuracy of 1 in 48,000. We conclude 
therefore that we may expect good results from most of these fornndae. But some 
remarks on their relative goodness may be of service. In the first place the Euler- 
Maclaurin formulae (a) and (/B) with four differences are not nearly as good as 
Mr Sheppard's new formulae (<.) and (/x) using only three differences, and not 
so good as (^) or (tt) with two differences. It seems to me accordingly that unless 
we are prepared to go to great labour and calculate high differences, (i), (/a), (^) or 
(tt) are the best formulae to use, and that for nearly all practical purposes (6) and 
(X) are quite accurate enough for use. Boole's Rule (e) and Weddle's Rule (^) give 
splendid results, but great care must be taken when we apply them to somewhat 
irregular observations of physical quantities and to frequencies, and not to the 
evaluation of mathematical integrals, for in the bulk of the formula? they weight 
and largely weight certain ordinates, and thus may tend to emphasise errors in 
particular observations. 
(3) It seems well to illustrate the application of these formulae to a special 
case, although in doing so I anticipate some of the results to be reached later. Let 
us try and fit by the method of moments a parabola of the third order to the 
following data : 
a; = 0 
y= -382 
X = -6 
= 1-270 
1 
•674 
•7 
1-215 
2 
-923 
•8 
1-137 
3 
1104 
-9 
•989 
4 
1-214 
10 
•819 
5 
1-273 
These data are really a series of measurements on Aneroid Barometers published 
by Dr Chree in a paper in the Phil. Trans., Vol. 191, A., p. 448. They will serve 
as well as any others, however, as an illustration of method. 
* Clearly: Area = 4j.- J (^y - nearly. This is a very useful formula — based on an assumption 
as to parabolic segments like Simpson's — when both extreme and mid-ordinates are available. 
