278 
On the Systematic Fitting of Curves 
Case (iv). Bounding ordinates with the two mid-ordinates only of the terminal 
elements. 
(a) One Difference: 
1 2t) 
Area = + g (^2p_l) ~ ~ ~ ^^-^^^ 
(6) Two Differences : 
. , 1 2jD (30o - 29) , , „ , 
Area = Ac + ^^ ( 2^,-1)0,-1 ) f^"^ " " " h 
1 2|)(10i>-9) „ X , f s 
- 120 (2p-B) (p-l) - - - "-'^^ 
If p be fairly large this is not very divergent from 
Area = Ac + i {{z^ - Zo) - {zp - Zp_^^\ h-^ {(z, - z0 - {Zp_>^ - Zp^,)} h (p), 
which may be obtained directly by a double application of Simpson's Formula, 
and is somewhat more exact than the lattei'. 
It is, perhaps, worth while exhibiting the sort of relative exactness to be 
obtained by the whole series of formulie on a special example, say I - — — for 
12 or 13 ordinates. We find 
1 dx 
0 1 + a; 
= -693,147,18. 
Divergence 
(a), with four differences, 
+ -000,000,25 
(/8), with four differences. 
- -000,000,59 
(7) 
+ -000,001,48 
(S) 
+ -000,003,28 
(e) 
+ -000,000,07 
+ 000,000,04 
iv) 
+ -000,014,59 
{6) 
+ -000,000,93 
(0 
+ -000,000,07 
(«) 
- -000,014,93 
(\) 
--000,001,26 
(/^) 
'- -000,000,12 
{v) 
- -000,003,91 
(^) 
- -000,000,14 
(o) 
+ -000,008,12 
(tt) 
+ -000,000,22 
ip) 
+ -000,001,27 
