318 METHODS OF INTERPOLATION. 



wliicb is identical with formula (1). It will bo found tbat the general 

 formula (11) can be obtained in this way more easily than in any other. 

 The particular feature of the first method of adjustment, that it makes 

 the arithmetical means of the terms in the corEespouding assumed 

 groups in the new series precisely equal to those in the original one, is 

 also characteristic of a method which has sometimes been employed in 

 solving equations of condition. (See the Calads Pratiques Appliques mix 

 Sciences d' Observation, by MM. Babiuet and Housel, page 81.) If the 

 law of a series is to be rei)resented by an equation of the form — 

 y=A. + B<p{x)+C'i'{x) + &G., 



where <p{x), y'{x), &c., do not contain any constants to be determined, 

 then there will subsist between any given terms or ordinates 2/1, 2/21 ^37 

 &c., and the corresponding abscissas Xi, X2, X3, &c., the following equa- 

 tions of condition : 



2/i=A+B^(^i) + C'/'(.ri) + &c. 



y2=A+B<f{x2) + Cii'{x2) + &c. 

 y2=A+B<p{X3)-^Cip{X3) + &c. 

 &c. &c. 



Let us suppose for example that there are only three constants, A, B, 

 and C, and that the number of terms in the given series is anj^ greater 

 number, for instance six. Then to reduce the six equations of condition 

 to only three, we may add them together in pairs or groups of two, and, 

 denoting the sums of the terms in the three groups by Si, S2, S3, we shall 

 have — 



Si=2A+B[^(a;,) + ^(.r2)] + C[v'(^i) + </'(^2)] 



S2=2A + B[^(j!^3) + f(*4)] + C[./'(.^3)+V^(^4)] 



S,=2A+B[<f{x,) + <p{x,)\ + C[iiix,)^il'{Xe)] 



Here there are only as many equations as there are constants to be de- 

 termined, and since Si, S^, S3, and Xi, Xz, &c., are known from the origi- 

 nal series, Ave can obtain the numerical values of the three constants. 

 Let these be A', B', and 0' 5 then the equation of the graduated 



series is — 



7/=A'+BV(.«) + C'v(^) 



and when the values a-i, .r2, .T3, &c., are successively assigned to the vari- 

 able in this equation, the resulting values of y will be the terms of the 

 graduated series, and the arithmetical means of the terms in the assumed 

 groups will be the same in it as in the original series. This will always 

 be the case, without regard to the number of terms in the series, or to 

 the number of constants and groups to be assumed, or to the extent or 

 position of the groups. It is not even necessary that the terms grouped 

 together sliould be consecutive, nor that the abs(!issas Xi, X2, x^, &c., 

 should be in arithmetical progression. 



This method, however, labors under certain disadvantages when com- 

 pared with the one which we have proposed. The computations it in- 

 volves are uuich more laborious, especially when the number of con- 



