Eleanor Pairman and Karl Pearson 
237 
Thus we can write 
where Cg-i is the " choixlal area" term and L^._i is tlie limit teiui of tlie Euier- 
Maclaurin series, or if = w,, + nJi , 
c,_, = h {^z,x,r' + z,xr' + z^xr' + • • • + kZy^rX 
Ls-i = h 
h' dHZx'-') ^ d^Zx'-'] 
1209600 dx-' 47900160 dn^ 
.(ix). 
We now turn to the evaluation of the chordal areas. We can obtain these by 
remembering that if be the frequency on the qth subrange h, 
= ydx = + + . . . + 
J x„ 
Thus it follows that 
7( = 1 
if we note that Z^ — Q. 
But the series coefficient of can itself be summed by the Euler-Maclaurin 
Theorem, i.e. 
h {\xr' + (.^-o + + • ■ • + (.n + (m - 1) h^'-^\ 
t'-'dt - yi{xo + iihy-' + 
J 
^dH^ 
dx-' 
= -ih (.To + uh.y-' + ^ (.r„ + ///O' + (.s - 1) ^2 (.?'o + 
- - 1) (.s - 2) (s - 3) ^i^A^ (.T„ + uhy-^ 
+ (.9 - 1 ) (,9 _ 2) (.s - 3) (.9 - 4) (.s - 5) ^jy^jjjh'^ (x„ + nh - . . . 
- x,^ -{s-l)~ xr"- + {s - 1) (.9 - 2) (,9 - 3) ^h^>'xr' 
_(,,_!) (s 2) (,9 - 3) (.9 - 4) (.9 - .5) + . . . . 
We are now in the position to find the value of sC',_, for the successive values 
of s. We have 
1 1 V 
^ (s6's_j) ,,=1 = ^ ^ \- yi + a^o + uh - .To} «H 
= Ty- >Sf {a?o + (?( - -J) /'} - 'tt^ 'S' ("«) = W--n (x), 
N 
