282 METHODS OF INTERPOLATION. 



Ave obtain tbe integral — 



S= r'^l''ydx=n[A+Bx-^C{x'-+j\n^) + Dx{x''+}n^)] . . (7) 



J x — in 



wiiicli expresses the sum S of any n terms taken in a group, tlie abscissa 

 of the middle point of the group being x. Substituting for n the four 

 Tiilues Wi, ih, W2j ni, in succession, and lor x the four corresponding val- 

 ue-s — «i, — rt^, +«25 aii<l -^(ii, we obtain the four ecpiations — 



Si = n, [A — Bffi+C {ar^j\,n,~y^ I>«i {a.^+W)] 



52 = n2 [A — B(<2+C («/ + yL«2') — ^«2 {a-i'+^n-r)] 



53 = % [A + Ba2+G (a,24--V»2'') + DosCo^^+im,^')] 

 S, = n, [A + Bai+C {a,^+j\n,') + 0% (ai^+im^)] 



These are sufficient to determine the four constants A, B, 0, D, and, 

 arranging (7) according to the powers of a;, we have — 



1 r %(12«r+7h^)(S2+S3)-n2(12«2^+»/)(Si+S,) \ 



1 /' ai9ti(4ai^+V)(S3-S2)-ff2^?2(4tf2^+%^)(S4-S]) ^ 

 'i»2\ 4:{ai^—a/)-\-n{^—)h^ 



"2(Sl+S4)-lh(S2+S3)\ >(8) 



C^ C /- »,(S,+S4)-ni(S2+S3) \ 

 niii2\ 12(ar— «2^)+»i'— "2''^ y 



■p^ 2 /^ q,3«3(S4-S0- ai7?i(S3-S2)\ 



«lrt2«l%\ 4(«.i^ — (<2") + «i'^ — ih'^ J 



S = n[(A+yLCM2) + (B+iD«2)x+C.r2+Da;3] 



This formula enables us to interpolate the sum S of any n terms in a 

 group i^recisely as (2) does, but more accurately. It gives exact results 

 when the series taken is of an order not higher than the third, and 

 approximate or adjusted ones in other cases. With anj' given series, 

 taking four groups of terms symmetrically situated on each side of a 

 middle point which become^s the origin of coordinates, we can construct 

 a new series of the third order, such that the aritlimetical means of the 

 terms in the four corresponding groups in it shall be equal to those in 

 the given series. If the four groups are consecutive and contain ni 

 terms each, we have — 



ai=^ni, «2=i»i 



and the constants reduce to — 



A=^^^[7(S2+S,)-(Si+S4)] 



B=j^,[15(S3-S2)-(S4-S,)] 



C=J^,[(Sl+S4)-(S2+S3)] 



(B) 



