310 METHODS OF INTERPOLATION. 



We have remarked tliat wbeu a series is graduated by means of 

 formulas sucb as (A), (B), (C), &c., tbe accuracy attained is greatest at 

 tbe middle of tbe series and least at its extremities. Tbe question tben 

 arises, wbetber tbe errors cannot be more equally distributed tbrougb- 

 out tbe whole series by making tbe number of terms in a group smaller 

 at the extremities and increasing up to tbe middle, instead of having 

 tbe number tbe same for all tbe groups. When any i)articular law of 

 increase is adopted, there will be no dilBculty in finding corresponding 

 formulas similar to (A), (B), &c., by which to compute tbe values of the 

 constants. For the results of some recent investigations by Tchebitcbeff 

 w'ith regard to the best arrangement of tbe data in making ordinary 

 interpolations, not from groups, but from single terms or ordinates, see 

 the Traite dc Calciil Differentiel of J. Bertrand, pages 512 to 521. These 

 naturally lead to tbe supposition that when tbe method of groups is 

 used, the best representation of a given series by another of algebraic 

 form will be obtained by regarding the whole interval which the series 

 occupies on tbe axis of X as being divided, not into equal portions, but 

 into portions which are the projections upon it of equal divisions of a 

 semicircle drawn upon that interval as a diameter, the number of these 

 divisions being made equal to the number of groups assumed. Of 

 course the number of terms in each grou[) will in general be fractional. 

 For a series of the second order, the numbers of terms in the thi'ee 

 assumed groups will be — 



,=»3=p^ri_cos|^=iN 



H2 = XcOS^=JiSr 



where N denotes tbe whole number of terms in tbe series, so that ^N is 

 tbe radius of the semicircle. In equation (1), 



S=w[A+B.r+C(^+ iV^2)j 

 we substitute for n its three values Wi, ^2, and n-i in succession, and for a; 

 tbe three corresponding values — 



07=— flS", x=0, a'=|]Sr 



thus obtaining the three equations of condition — 



S:=iX(A-fBN+J30K2) 



S2=iN(A+J^CN^) 



S3=iN(A+fBN+f^CN'^) 

 These determine A, B, and C; and arranging the original equation 

 according to the powers of ic, we have tbe fornuila — 



A=^[7S2-(Si+S3)] 



Wi: 



.B=l« 





