RELATIONS APPLICABLE TO INTERPOLATION II. For interpolation, where u is a continuously varying func tion of x, and values of u are tabulated at intervals h in x, the relations between differences of the u's and differential coefficients of u are important. The values of x for which u is tabulated are taken to be • - • • • , where If the differential coefficient of us with regard to x is denoted by Du, we may regard D as an operator; and it will be found that this can be combined with the operators A, E, etc., and with numbers, according to the laws of ordinary algebra. Taylor's Formula (see CALCULUS, DIFFERENTIAL AND INTEGRAL), Expanding and combining these, we get expressions for S, µS, etc., in terms of hD; and thence we deduce expressions for hDu, etc., in terms of the relevant central differences, original or con structed.
If in (15) we replace h by 6h, and f (x -f-ah) by we can write it in the form This formula enables us to interpolate on both sides of by using the central differences which are in a line with (The values of the c's are given more fully under INTERPOLATION.) There is a formula of the same kind for interpolating between x = xo and x= in terms of and the central differences which are in a line with it.
V. RELATIONS APPLICABLE TO QUADRATURE 12. Just as under the heading Interpolation we deal with the relations between differential coefficients and differences, so under Quadrature we deal with the relations between integrals and sums. The main problem is that of expressing the area of a figure (of the kind with which we are familiar in dealing with graphs) in terms of selected ordinates of the figure. We can, however, reverse the process, and express the sum of a series, accurately or approximately, in terms of an integral. The im portant theorem is the Euler-Maclaurin theorem, given under