276 On the Systematic Fitting of Curves 
Case (i). Bounding ordinates or chordal area known. 
(a) One Difference : 
1 p 
Area = ^ c + j2 {p -l) ~ ~ ~ ^^-^^^ 
If we take p/(p — V) to be approximately unity, this formula reduces to (a) 
retaining only the first difference. 
(b) Two Differences : 
Area = + -^^Ifc??! - z.) - (z, - .,_,)) * 
(c) Differences : 
Area-^c + 5()4Q - 1) - 2) - 3) 
_ j)(llV-504;> + 432) _ 
1260 - 2) - 3) - 4) '^-"^^ " 
1 j.(133;.'-462^ + 360) _ 
+ 5040(^:>-3)(;>-4)(p-5)^^^^ ^^^-^ 
Case (ii). Mid-ordinates or tangential area known. 
(a) One Difference : 
1 p 
Area = ^ r - 24 ^^^^ ~ ~ ~ ^^'-^^^ 
(b) Ttoo Differences : 
Area = ^ 7- yg^ (^_2)(^-3) ^^^^ " ' ^'^'^ " ^^"^^^ ^ 
1 « (40» - 57) , , , ,^ , 
+ 960 ( ^-3)(j.-4) ^^'^ - - ^'--^ - '^-^^^ ^ 
This formula has many advantages, it is more exact than (k), and although less 
so than (fi) is sufficient for most practical purposes. It weights in the bulk of the 
formula, A^, all the ordinates equally and thus is superior to those of Case (i) 
which give only half-weight to the terminal ordinates. In order to facilitate its 
use, writing it in the form 
Area = At-P {(^i - ^i) - (zp-^ - Zp-^)] h + Q {{z^ - z^) - {Zp_^ - Zp_i)} h, 
