Transactions of The Royal Society of Canada 



SECTION III 



Series III 



MARCH 1916 



Vol. IX 



An application of the "Calculus of Finite Differences'' to correct an 



experimental curve, and thus obtain, by a graphical method, an 



accurate representation of the integral of this curve. 



By S. Douglas Killam. 



Presented by H. M. Tory, F.R.S.C. 



(Read May Meeting, 1915). 



Let us suppose that as the result of an experiment we obtain 

 n values of a function corresponding to n values at equal intervals of 

 an independent variable x. In the example used to make the problem 

 more concrete I have taken n equal to 6, and have the following table 

 of values : 



Our problem is to obtain a graphical representation of the func- 

 tion f{x), a polynomial of the fifth degree passing through these 

 six points in an x ^'-plane, and from that a graphical representation 



of the function 



f{x)dx. 



If we solve this problem analytically we obtain a polynomial 

 of the fifth degree passing through the six given points, by Lagrange's 

 interpolation formula 



(« — *2) {X — Xz) {X-Xi) {X-Xi) {X-X^) 



/(^)=r^ 



/(:^i) + 



+ 



(:k:i-X2) {X1-X3) {xi-Xi) {xi-Xi) ixi-Xi) 



, _ _ {X-Xi) {X — Xj) {X-X3) jx-Xj) (x — Xb) ., N 



"^ ' '^ {X^ — Xi) (,X(, — X-i) {Xf, — Xz) {Xi — Xi) {Xf,-Xi) •'^ ^' 



where (xi, yi) ; {x^, y^) ■ ■ ■ ■ (^e ye), are the co-ordinates of the six points. 



Sec. Ill, 1916—15 



