METHODS OF INTERPOLATION. 319 



slants or tlie number of terms iu the series is large ; it does not give any- 

 general expression like (12) or (44) for the sum S of any n terms taken 

 in a group, and it does not permit the use of groups composed of a frac- 

 tional number of terms. 



ADJUSTMENT OP A DOUBLE SERIES. 



By methods entirely analogous to those which have been applied to 

 functions of one variable, we can proceed to graduate an irregular dou- 

 ble series or table of values of a function of two variables. The table 

 is supposed to be arranged in the usual rectangular form, the successive 

 values of each variable being equidistant. Tbe intervals between any 

 tw^o such values, however, are not necessarily the same for both varia- 

 bles. The algebraic equation — 



is the equation of a curved surface. The rectangular table being sup- 

 posed to be situated in the plane of X Y, with its sides i>arallel to the 

 axes of X and Y, and its middle point coinciding with the origin of co- 

 ordinates, let a series of equidistant vertical planes be drawn parallel 

 to the i>lane of Z Y, and another series of planes iu like manner parallel 

 to the plane of Z X, so that the intersections of these planes with the 

 plane of X Y shall form the divisions of the given table. Each of these 

 divisions is the base of a solid which is limited at the sides by the ver- 

 tical planes and at the top by the curved surface. Every such solid 

 may be regarded as representing the corresponding tabulated value of 

 the function, and the sides of the bases are taken as unity, but the units 

 lying in the directions of x and y are not necessarily equal to each 

 other. If we assume a group of adjacent divisions of the table, situated 

 so as to form a rectangle whose sides, parallel to the axes of X and Y, 

 consist each of m and n units respectively, then the solid included be- 

 tween this rectangular base, its limiting vertical planes, and the curved 

 surface, will be rej^resented by the integral — 



' dy I zdx 



y'—in tJx'-im 



where x' and y' are the coordinates of the middle point of the rectan- 

 gular base. Performing the integrations indicated, and omitting the 

 accents from x' and y', we have — 



B=mn[A-\-Bx-\-Cy-{-B{.v'+-^\m^)-^F4y^ + j\n^) + Fxy-]-&c.] . . . (46) 



This solid is evidently the sum of the solids which belong to the 

 several divisions of the assumed groui), so that the formula expresses 

 the sum S of the terms iu any rectangular group in the table by means 

 of the numbers m and n of terms contained in each one of tlie sides of 

 the group lying parallel to the axes of X and Y respectively, the coordi- 

 nates X and y of the middle point of the group, and the constants A, B, 

 Q, &c. For any group assumed we know the numerical values of S, m, 



