CALCULUS OF DIFFERENCES (Theory of Finite Dif ferences), that branch of mathematics which deals with the suc cessive differences of the terms of a series.
1. The subject falls under four main heads.
(i.) As a simple example, take the series which is the series of squares of the positive integers. If from each term of the series we subtract the preceding term, we get a new series which is the series of odd positive integers. The terms of these series are the first differences of the terms of the original series. By a second set of subtractions we get the second differences, each of which is 2; and so on. We can put these in tabular form, thus: The form of this series suggests that the sum of n terms of (A) is and this can be verified by mathematical induction (q.v.).
(ii.) We can express the above in a slightly different way by saying that there is a certain unknown series (S), which we want to find, and that our data are the first differences of this series, given in (A). Or, more briefly, we can say that the difference be tween the nth term of (S) and the (n+ I)th term is 2n+ r. This is an example of a difference-equation (§Io). In the notation explained below, the equation would be stated in the form 2n+ 1, the quantity to be found being (iii.) In the class of cases considered above we are concerned with a discontinuous series of terms, and we are limited to the consideration of this series or of the series derived from it by processes of differencing or summing. There is another class of cases in which our data are values of a continuously varying quantity, and our first object is to find intermediate values of this quantity. This process is interpolation (q.v.), which leads on to quadrature (q.v.). The importance of the calculus of differences in relation to interpolation and quadrature arises from the fact that, if p is a positive integer, the difference between nP and (n+ I )p is an expression in which the highest power of n is so that, if n is a polynomial in x, the values of which are known for a series of integral values of x, the process of successive dif ferencing of these values leads ultimately to a series of differences, all of which are o.
(iv.) Underlying all these processes there are certain relations between terms and differences, which it is convenient to deal with first. The subject will accordingly be considered, though very briefly, under the heads (I.) Algebra of differences and sums; (II.) Sums of series; (III.) Difference-equations generally; (IV.) Relations applicable to interpolation; (V.) Relations appli cable to quadrature. Applications of (IV.) and (V.) will be found under INTERPOLATION and MENSURATION respectively.