METHODS OP^ INTERPOLATION. 315 



considered, aud let a be tbe value of x correspouding to the first term ; 

 tUeu the sum of the terms in the gi-oup is — 



s=^i«+j)-/i(«-i)-h/i«+§)-%+i)-K/i:«+f)^\«+t) 

 + +/("+«-*)-/(«+«-§) 



which cancels at once to — 



S=/(a+n-^)-/(«-i) 



Now, if x' be the value of x correspoiidiug" to the middle of the group, 



we have — 



iv'=a + ^{n — l) 

 aud cousequeutly — 



a=x'—hi-[-i 



so that the expression for S reduces to — 



^=Jlx'+hi)-f{x'-hi) . . . (44) 

 We can conceive that, by varying the form of the function / aud the 

 values of the constants which it contains, the series of values of u cau 

 be made to approximate more or less closely to any given series of equi- 

 distant numbers which follow some general Law-. Hence, to graduate 

 such a given series, we have only to assume a function f{x) of suitable 

 form, aud substituting in it first x-]-hi and then x — hi in place of the 

 variable x, the difference between the two results will express the sum 

 S of the terms in any group in the graduated series b^^ means of the 

 number n of terms which that group contains, the abscissa ic? of the mid- 

 dle point of the group referred to an assumed origin of coordinates, and 

 the constants which are involved in the function f{x). In the case of 

 any single group the values of n aud x are known, and the value of S 

 being taken equal to the sum of the terms in the corresponding grouj) 

 in the given series, we shall have an equation of condition containing 

 only the unknown constants aud numerical quantities. By assuming 

 as many groups as there are constants, we obtain a number of equations 

 just sufficient to determine the values of the constants. Substituting 

 these values in formula (43), we obtain the equation which expresses 

 the empirical law of the given series, and from which the graduated one 

 may be constructed. The arithmetical means of the terms in the 

 assumed groups in the graduated series will be severally equal to those 

 of the terms in the corresponding groups in the given one. 



If we assume more groups than there are constants, there will result 

 a number of equations of condition greater than the number of con- 

 stants to be determined. The values of the constants can then be found 

 by the method of least squares. In this way we may expect, in certain 

 cases, to increase a little the degree of general accuracy with which the 

 graduated series represents the given one, without at the same time 

 increasing the number of constants and raising the degree of the equa- 

 tion. But of course the arithmetical means of the terms in the cor- 

 responding groups in the two series will now be only approximately 



