198 THE ROYAL SOCIETY OF CANADA 



Teubner.] Our integral curve f(x)dx obtained in this way is 



^^ . . . 

 shown with its graphical construction in Figure 2. If our graphical 



work is correct the curve o b' c' d' e' f is an accurate graphical repre- 

 sentation of the polynomial 



* f{x)dx= 7^ W -3x^+5a;4+5ic3+49ic2+360ic 

 120 1^ 3 



If we plot this curve, it coincides exactly with our graphically found 

 curve. The dotted curve between e' and k shows the error in our 

 integral curve when we drew a smooth curve through the six points 

 a, h, c, d, e and/, as representing /(^c) without extrapolating the points 

 r and g. The error in my example is very small ; but would have been 

 much greater if there had been a correction to f{x) between a and h. 

 Instead of the above used graphical method of integration we might 

 have integrated our curve /(a:) by the "integraph" of Coradi; but I 

 have found by long practice that the graphical method is more accurate 

 and solves the problem in less time. 



The graphically found curve 



f{x)dx is within the accuracy of 







graphical methods, and can be found without the long process of 

 applying Lagrange's interpolation formula, and of plotting the integral 

 curve after integrating analytically. In short, we proceed by a 

 graphical method or construction directly from our experimental 



r* 



curve to the desired function f{x)dx. 



JO 



