so that from a geometrical point of view yo, Y1, Y2, Y3, equidistant ordinates of the curve y = F(x). Now these quantities ▲ F(x), A2F(x),..... are called the successive differences of F(x); and in all cases to which the preceding formula of approximation can be successfully applied, it is necessary that the values of these differences should rapidly decrease; this circumstance may easily be verified, and it is found to be true in all cases of tabulated functions. Thus, if F(x) is a rational algebraical function of n dimensions, ▲* F(x) = 0. On this hypothesis then we shall deduce from (213) successive approximate values to the definite integral. Thus, if we omit all terms after A2F(x), we have From (216) may be deduced a rule, originally discovered by Thomas Simpson, and frequently employed in mensuration. n Let the range x-x。 be divided into n equal parts, where n is an even number, each of which = i; then as (216) gives the value of the integral for any two consecutive parts, taking the sum of all, we shall have, i [**¥(x) dx = {} {Yo+4Y1+ Y2+Ya+4 Ys+Ys + ... 3 i = +2(Y2+Y4+...+Yn_2)}. (218) 3 {Yo+Y+4(Y2+Y3+ ... + Yn−1) Again, if all terms in (213) after A3 F(x) are omitted, we have a x2, (220) and If ▲3F(x) = 0, as is the case when F(x) = (221) give results strictly accurate. 115.] An approximation may also be made by the following process, which assigns certain limits within which the true value of the definite integral lies. Let F'(x) be the element-function of the definite integral; and let f'(x) and p′(x) be two other functions of x, the values of which are respectively greater and less than F(x), for every value of x within the range of integration; so that for every such value ƒ'(x) > F′(x) > p′(x); Moreover let us take ƒ'(x) and '(x), such that the definite integrals, of which they are the element-functions, may be determined: then evidently, by the definition of a definite integral, [**ƒ^"(x) dr >["¥ (x) dx > [′′ 4(x) dæ. (222) Let us investigate by this method an approximate value of the so that limits are assigned very near together within which the value of the given definite integral lies. 116.] The definition given in (200) suggests other means of obtaining approximate values of definite integrals, and of ascertaining certain properties relating to those values. Let F'(x) be the element-function: then if a, is greater than , and F(x) does not change sign within the range of integration, [**F′(x)dæ has the same sign as r'(@). If however r'(2) changes sign, the integral will be positive or negative, according as the positive or negative part of the series is the greater. Since, in the series on the right hand member of (200), x-xo, XX1 are all quantities of the same sign, by preliminary Theorem III of Vol. I. the sum of the series is equal to the product of the sum of all these quantities and some mean value of the other factor: consequently [**r'(x)dx = (x„—x.)¥′{xo+0 (x„−xo)}, (224) wherein denotes an undetermined proper positive fraction. Then, the limits of the value of the definite integral, given by substituting 0 and 1 severally for 0, are (x-x) F′(x) and (x) F(x), between which the value of the definite integral lies. n The geometrical interpretation of (221) deserves a passing notice. Let y = F(x) be the equation to the curve P, PQP, in fig. 16; Then as the definite integral expresses the area Po Mn Mo and as Mo Max-xo, the equation shews that the area is equal to that of a rectangle one of whose sides is MM, and the other is an ordinate to the curve corresponding to an abscissa intermediate Mn If the left-hand member of (200) is determined by means of its indefinite integral; then F(x) F(x) = (xn−xo) F′ { x。+0 (xn− xo)}, which equation has been already found in Art. 111, Vol. I. If the range a,,-x, is infinitesimal, - 'F'(x) dx = (x-xo) F′(o); (225) (226) which is the first term of the series, of which the definite integral is the sum. If the element-function F(x) is the product of two functions, f(x) and f'(x), of which p'(a) has the same sign for all values of æ within the range of integration; then [**ƒ (x)4′(x) dx = (x ̧−x ̧) ƒ (x ̧) $′(x) + (x2−x ̧) ƒ (xg) $′(xg) + ..... ...+(x-x1) '(x-1). (227) Now the right-hand member of this equation consists of a series of terms, into each of which a factor enters, which is of the form (x-x;) '(x), and is always of the same sign; therefore by Preliminary Theorem III, Vol. I., = ƒ {xo+0 (xn−Xo)} {(X1—X。) p′(x) + (x2− x1) ¢' (~1) + ... Equation (224) is evidently a particular case of this theorem. If the indefinite integral of the right-hand member can be determined, this equation becomes [**ƒ (x) 4′(x) dx = ƒ {x。+0 (x„−x ̧) } { $ (x„) — $ (x) } • (229) Hence it follows that if G and L are respectively the greatest and the least values of f (x) within the range of integration, xn c [**4′(x) dx > [""ƒ (x) 4′(x)dæ >1. [**4′(x) dx. G (231) These theorems are frequently of considerable use in determining the limits of value of a definite integral; and also in shewing the value which a definite integral takes for a particular value of a constant contained in its element-function. Thus we may dedx termine the value of when a = x. On compar e-ax of / 0 ing this with the formula given in (228), let f(x) and = when x = 0, so that the mean 2 between these quantities. Also e-a -ax2 dx = 117.] Of the various processes which have been devised for the purpose of approximating to the value of definite integrals, some consist in the development of the element-function into a series, and in the subsequent integration of each term separately; in others, series are formed, either by integration by parts, or by some other method, in general terms, and these are applied to the particular element-function. The following series was devised by John Bernoulli. Let (**r' (.x) da be the definite integral whose value is required : ΤΟ This theorem may be proved, and also its general term may be found, by means of Taylor's series; and hereby the limits of the value will be determined. so that replacing F(x) by fr′(x) dæ, and taking the definite inte gral of both members, we have |