308 METHODS OF INTERPOLATION. 



plienomenon so far as that law can be expressed by an algebraic and 

 entire function.* 



In practice, when this method is to be applied to the graduation of 

 a particular series, it will not be essential to have the assumed groups 

 contain an equal number of terms each, nor to make the groups consecu- 

 tive. Their iiositions, and the number of terms they contain, may be 

 entirely arbitrary. The integral — 



S 



Jx—\n 



expresses the sum S of the terms in any group in a series of the ?wth 

 order by means of the w + 1 constants A, B, C, «&c., the number n of 

 terms which the group contains, and the abscissa x of the middle point 

 of the group, each term in the series being regarded as an area occupy- 

 ing, on the axis of X, a space equal to unity. In the case of any one of 

 the assumed groups, we know the sum S of the terms in it, and their 

 number ??, and the abscissa x of their middle point, so that we have an 

 equation of condition which, besides the m+1 constants A, B, C, «S:c., 

 contains only numerical quantities. Each group assumed furnishes one 

 such equation. By assuming w + 1 gi'oups we shall have as many equa- 

 tions as there are constants A, B, 0, &c., to be determined, and hence 

 it will always be possible to find the numerical values of the constants. 

 Substituting their values in the general expression for S, arranging the 

 terms according to the powers of a?, andi)uttingw=l and S=w, we shall 

 have an equation of the form — 



u-A'-\-Wx-\-Q'x^+ +T'.r'° 



which will be the equation of the graduated series, and from which that 

 series may be constructed. It will have its mth differences constant 

 and the arithmetical means of the terms in the corresponding groups 

 in it will be severally equal to those of the terms in the m + 1 groups 

 assumed in the original series. 



But although the positions of the groups and the numbers of terms 

 which they may contain are thus unlimited in theory, it will probably 

 be best in most cases to make them consecutive and consisting each of 

 the same number of terms. When the law of a series varies very rap- 

 idly in some places, and slowly in others, it may indeed be necessary to 

 assume, at those portions of the series where the variation is most rapid, 

 a larger number of groups, consisting of fewer terms each, than will be 

 required in the portions where the variation is slow. But with a fixed 

 number of groui)s, the process of finding the values of the constants A, 

 B, C, &c., will be simplified if the groups are assumed so as to be sym- 

 metrically situated on either side of the origin of coordinates ; that is, 

 situated in such manner that for every group of terms whose abscissa 



* The constant difl'ereiice of the abscissas or argnments is here assnmefl to be unity. 

 But if we wisli to rej^ard it as any other quantity h, we shall merely have to substitute, 



in the final equation, ,- in the idace of x. • 



