284 
On the Stjstematic Fitting of Curves 
and for = 0 we have 
dZ 
daf 
where a.^, a.^, ... as, ... define the contact of the integral curve at the origin with 
the line Z = N, and will be supposed for the time being known. 
The frequency curve and its integral curve will accordingly take the form 
indicated in the diagram below. 
a; = 0 
Now by the Euler-Maclaurin formula 
f z'dx = (\z: + Z,' + Z,' + ... + Z'i_, + \Zl) h 
J 0 
VBJ^dZ^_B^d^ BJld'Z^ _ 
The expression in square brackets vanishes at the upper limit not only for Z' = Z, 
but for Z' = Zx^, — since every differential containing either Z or one of its differ- 
ential coefficients 
At the lower limit we have by applying Leibnitz's Theorem 
'dTU^af) 
dx"" 
= ?i(71-1)(h-2)... ( + 
dT^Z 
daf^-' 
\n — s 
provided n be greater than s, otherwise it is zero, unless n=s, when we have the 
value [n N. 
Now let Gsh stand for the chordal area. 
(i^o-V + Z,x,' + Z,x.,' +...+^Zil') h, 
