METHODS OF INTERPOLATION. 277 



polatiou. It is not witliout interest when regarded from a purely 

 matliematical point of view. The general question as to liow an 

 irregular series can be made regular is answered by means of the 

 obvious principle that, altliougli single terms in a series may deviate 

 considerably from the normal standard, yet the arithmetical means of 

 successive groups of terms will be less iiuctuating, because the errors 

 of the single terms which compose each group tend to compensate each 

 other, and also because the means of two groups which are partly com- 

 posed of the same terms must necessarily approximate toward each 

 other as the number of terms common to both is increased. In ordinary 

 interpolation, we proceed from some known single terms in a series to 

 find the values of other terms ; in the present case, however, all single 

 terms are unreliable, and the problem is to determine the single terms in 

 a series when only the arithmetical means of some groups of terms are 

 given. To find expressions for the sum, and consequently the mean, of 

 the terms in any group, we shall make use of the known principle that, in 

 a continuous series whose law is given or assumed, the sum of a limited 

 number of terms can be regarded as a definite integral, which is the 

 aggregate of a succession of similar integrals corresponding to the terms 

 considered.* 



FIRST METHOD OF ADJUSTMENT. 



We know that when equidistant ordinates are drawn to the jiarabola — 



they form a series of the second order ; that is, their second differences 

 are constant. Let c represent the distance from one ordinate to another j 

 the area of the curve included between two such ordinates will be — 



^^ dx = c[K + Bo;' + C(if'2 + tV <^)\ 



■vrhere x' is the abscissa corresponding to the middle ordinate of the 

 area. Since this area is a function of the second degree in x'^ it follows 

 that when values in arithmetical progression, such as 1, 2, 3, &c., are 

 assigued to x'^ the resulting areas will form a series of the second order. 

 This being premised, let us assume any three areas, Sj, S2, S3, so situ- 

 ated that the middle ordinates of Si and S3 shall fall respectively to the 

 left and right of the middle ordinate of S2, which is taken as the axis of 

 Y. Let 7ii, 7i2, ns, be the portions of the axis of X which form the bases 

 of these areas, and let «i and (h be the portions of the same axis inter- 

 cepted between the axis of Y and the middle ordinates of Si and S3 

 respectively. Then we have — 



Si=: r"''''^^"V<7^=Mi[A-Brti-fC(ai^-f,V'i2)] 



J -ai — \ni 



82= r^%jdx=n,{K-^i^iH') 



J —irii 



* See a note by M. Prouhet, appended to Vol. II of Sturm's Coins d' Analyse de VlScole 

 Folytechiique. 



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