196 THE ROYAL SOCIETY OF CANADA 



The polynomial of the 5th degree through the six points (0, 3), 

 (1,4) (5, 5) obtained by Lagrange's method is 



fix) = j^ (2x5-15^4+20x3 + 15x2 + 98x+ 36o} (1) 



We can integrate this function analytically and have 



f{x)dx = :^<~-3x' + 5x'-\-5x' +49x^+360x1 (2) 



where 



f(x)dx = o when x = o. 



This last function can be represented graphically in an x 3'-plane ! 

 but in obtaining it we have resorted to an intermediate analytical 

 step involving the use of Lagrange's formula, which is a long, tedious 

 piece of work in which the probability of error is very great. 



The alternative method is purely graphical and passes directly 

 from the given data by a graphical construction to a graphical repre- 



sentation of the integral curve 



f{x)dx. The accuracy of the 



graphical method is great enough for most problems in applied mathe- 

 matics; that is, the error is less than the width of the narrowest con- 

 struction lines possible to use. 



We begin (Fig. 1) by drawing a smooth curve through the six 

 given points a, b, c, d, e and f; and assume this to be a polynomial 

 of the fifth degree. If we compare this freehand curve with equation 

 (1) we find a shght error between e and f. If we knew the equation 

 of the curve /(x) we could easily plot other points and correct our 

 curve; but my object is to avoid the labor of finding this equation. 

 The method of extrapolation by finite differences is most useful for 

 my purpose. As /(x) is a polynomial of the fifth degree the fifth 

 column of differences is constant, and in my example equal to 2. (See 

 Table I). 



