OF INTERPOLATION AND EXTERMINATION. 341 



— 2_o^^24w —32m +5^m —4m 



12 3 4/ 



= l(-U^+itu -^f^U +XfM +J^M \, which is 

 ^ 12 3 4/ 



the area beyond the rectangle lu, and adding this as the 

 correction of the fluent, we have for the true area -^U^u-\- 

 32?/ -fl2M -f32M +7m ). This interpolation is very ac- 



1 2 3 4^^ 



curate where the curve does not become extremely oblique 

 to the absciss: but for a semicircle, or seuiiellipsis, it gives 

 the area too small in the ratio of .7737 to .7854, and if 

 great accuracy were required in £^ similar case, it would be 

 proper to divide the curve into two parts, and to compute 

 the area of each separately : or to add a little by estima- 

 tion ; to take, for example, 8m instead of7M, which would 

 make the area of the semicircle .784. 



Scholium 3. If the ordinates are not equidistant, it 

 will be easiest to represent them by an equation of the 

 form y—a-\'hx-{-cx^ + dx^ + • • . consisting of as many terms 

 as we have values of y, and finding each of the unknown 

 quantities a, 5, c, . . . , by comparing these values with each 

 other. This process is generally a little tedious, and it is 

 not possible to shorten it materially by any artifice, though 

 the results may be expressed in a form which is not wholly 

 without symmetry. 



388. Theorem. If there be any number 

 of linear equations, involving as many un- 

 known quantities, in the form a a:-{-b y-\-. , . = 



1 1 



A , a x+b ?/+. . ,—A , . . . ; we shall have i= 



12 2 2 



aA — ^A -\-yA — . . . 



-; the coefficients «, ^, y, be- 



2 3 



aa — ^a +ya 



