[ eillam] 



CALCULUS OF FINITE DIFFERENCES 

 Table I. 



197 



By starting with the fifth column of differences we can work back- 

 wards and find the values of/ (x) corresponding tox=— 2;x=— 1; 

 and X = 6. In my example it is only necessary to locate the points 

 (6, 16), and (— 1, 2) in order to draw a smooth curve through the 

 eight points r, a, b, c, d, e,/ and g; so that between a and/ this curve 

 is an exact graphical representation of the function /(x). In Figure 

 1 the dotted curve shows the correct shape of our function between 

 e and /. The part es'f was obtained before we found the point 

 g and is incorrect by the amount shown in the figure. The same 

 thing applies to that part of our curve between a and b, only in the 

 example taken the error is almost negligible. 



Now that we have a correct graphical representation of the func- 



tion fix) we wish a graphical representation of the function 



f{x)dx. 



The method of graphical integration first suggested by Massau^ 

 is the shortest and most accurate for our purpose. We replace the 

 curve fix) which is to be integrated (Figure 2) by a step-curve such 

 that the area under the step-curve equals the area under /(^c). We 

 then integrate the step-curve graphically, which is very simple when 

 we notice that the step-curve is made up of lines parallel to the x- 

 axis. [For literature on this method see Runge "Graphical Methods" 

 § 13 Columbia Univ. Press; or von Sanden "Praktische Analysis" 



1 Massau, "Mémoire sur l'intégration graphique et ses applications." 



