K. Pearson 
275 
Newton's Rule (3p elements), 
I zdx = lh{Zo + ^{zi + + Zi + Zs + Zt + Zi + ...) + z-ip 
+ 2(^3 + ^6+ ••• +^3p-3)} (^)- 
Boole's Rule* (4p elements), 
I zdx = {7zo + I4<(z, + Zs+ ... + z,p-i) + 7z,p 
+ S2{z^ + Z3 + z^+ ... + z,p-i) 
+ 12(Z2 + Ze + Zio+ ■■■+Zip-2)] (e)- 
Weddle's Rule {6p elements), 
'6P 
zdx = j%h {Zo + Z.;i + Zt+ Zs + Z^o + ■■■ + ^6p-2 + ^ap 
+ 2 + + • • • + ^6p—b) 
+ O {Zi + Zs + Z7 + ... + Zep-i) 
+ Q(z3 + Zs + z,5+ ... + z^p-i)} (?)• 
All these rules give with increasing exactness the value of the integral, but 
they suffer under obvious disadvantages : 
(a) The number of elements cannot often be selected beforehand, and if for 
example there be 7 or 11 or 13 a new rule has at once to be worked out. 
(b) The multiplying different ordinates by different factors is a source 
fruitful of arithmetical slips. 
(c) The multiplying of certain ordinates by factors much larger than others, 
multiplies the error made in the determination of certain ordinates largely. We 
do not give equal weight to all the ordinates. 
Thus, while formulae like (e) or (^) give extremely good results, especially for 
the integration of continuous mathematical functions, and this with less work than 
(a) or (fi). they do not seem advantageous for what we may term observation-curves. 
Accordingly Mr Sheppard has determined f the best coefficients for the corrections 
to the chordal and tangential areas when one, two or three differences only are 
used. He has provided the following quadrature formulae which seem to me of 
much interest and practical value. 
* I do not know who originated this rule ; it is given in Boole's Finite Differences. 
+ Mr Sheppard, since this memoir was written, has given the proofs of his formulae, L. Math. Soc, 
Proc, Vol. 32, p. 270. 
