K. Pearson 
273 
If I venture here to deal at some length with quadrature formulae, it is because 
the choice of a good formula is essential to the application of the method of 
moments. At the same time, although much will be familiar, there are new and 
novel points to which I want to draw special attention. For this portion of my 
paper I am chiefly indebted to the kindness of Mr W. F. Sheppard of Trinity 
College, Cambridge. I told hira that I wanted the best correctional terms for 
the tangential and chordal areas, and the working-out of the system of formulse 
is entirely due to him. 
An area may be looked at as given in two ways : (i) by extreme ordinates, or 
(ii) by mid-ordinates. The former we will represent by Zq, z^, z^, z^, ... Zp and the 
latter by z^, z^, z^, ... Zp_^, Zp_^. These ordinates will be supposed taken at equal 
distances, h, and for the purposes of practical calculation, h can nearly always be 
taken as our horizontal unit. We have thus the two schemes : 
For these cases we have respectively the Euler-Maclaurin formulae 
P zdx = Ac + h {yA - 72^' + ys^'-yA' + ••■)(^o + Zp) («), 
'zdx = AT-h (7/A - 7;A= + 7/ A' - + ...) + z^_^) (j3). 
Here Ac = h(^Zo + z^ + Z2+ ... + Zp_-, + ^Zp), 
and AT = h{zi^ + z^-it- z^ + ... - Zp_i-it- Zp_^), 
are respectively the areas of the chordal and tangential series of trapezia. Thus 
the formulae (a) and (/3) give the corrections which are respectively to be added 
and subtracted from what we may term for brevity the chordal and tangential 
areas in order to obtain the curved area. 
In the above formulae A operating on Zp and Zp^)^ must be taken backwards, i.e. 
Azp = Zp__^ — Zp and £^Zp-i = Zp_^ — Zp^^, while 
A^„ = 2'i-2r„, A^-j = ^|-^i. 
Biometrika i 27 
