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In this case we may take (97), and expand in a series of sines; and we have

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The geometrical interpretation is as follows. The locus consists of three straight lines, wbich, with the x-axis, bound a trapezium, the first and third being inclined to the x-axis at 45° and 135° respectively; and the second line being parallel to that axis at a distance a from it; the base of the trapezium on the x-axis = 1, and the length of the opposite side = 7—2a. On the negative side of the origin is an equal trapezium, which is below the axis of x. And these two trapeziums are repeated continually along the x-axis. If a = 5, the trapeziums become isosceles triangles, and

f(0) = { sin « – 22 sin 3 x + 3 sin 5x –... (141) 204.] Thus far we have assumed the limits of the definite integral (99) to be finite quantities ; let us now consider the form which that equation takes when c= 00. And to avoid all difficulty let us assume that f(x) never = oo, and = 0 when x = -2 and when x = 0o. From (99) we have f(x) = 2. Sof(z)dz +22=1}* (z)cos nT (7 — 2) dz. (142)

Now I f(z)dz = 0, when c = cc, under the conditions to which z is subject; so that the value of f(r) is given by the second term of (142). In t is let "* = ; then as n is increased by 1, u is increased by *; which is infinitesimal, when c = , and so must be denoted by du; and the summation becomes integration, the range of integration being the same as that of the summation; and thus the range of u extends from 0 to oc; hence we have

f(x) = /" f(z)cosu (x –z)dz du. (143)

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This theorem is called Fourier's Theorem *; and is of great importance in higher analytical investigations. If the limits of the u-integration are – oo and co, then

f(z) = LS 8(z) cosu (x 2)dz du. (144) If f (x) = f(-x), then

(* $(2) cos uz dz = 2 S(z)cos uz dz, Sof(2) sin uz dz = 0; therefore from (143),

f(x) = f(z)cos ux cos uz dz du. (145) This theorem, as (100) shews, is true, when x is equal to either of the limits of the z-integration. If f (x) = -f(-x), then by a similar process,

f(x) = [TF(z) sin ux sin uz dz du. (146) This theorem, as (101) shews, is not true, when x is equal to either of the limits of the z-integration.

205.] The applications of these theorems are very numerous, and the values of many definite integrals may be determined by them. For it is evident that if either one or the other of the integrations can be effected, the value of the remaining definite integral will also be determined. The theorem may also be applied to problems similar to those of Arts. 202, 203.

Ex. 1. Find a function of x which = 1 for all values of x between - 1 and 1, and = 0 for all other values. Here f(x) = f(-x), and we have evidently from (100),

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Il cos ux cos uz dz du 2 p cos ux sin u ,

(147) Tr Jou Ex. 2. Let f(x)= e-ax from x = 0 to x =00; and f(x) = pax from x = -00 to x = 0; then f(x) satisfies the conditions required for the theorem, and also those in (145); and we have

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"Jo q2 + 22" * See Théorie Analytique de la Chaleur; par M. Fourier, pp. 431, 445.

po cos ux du
ii 24 m2 = 2a e-";

(148) which result is the same as (121), Art. 101.

Ex. 3. Let f (x) = sin ax, from x = 0 to x = 0; and f (x) = sin (-ax), from x =-00 to x = 0; then f (x) satisfies the conditions required in (145); and we have

2 popoo
sin ax = | sin az cos uz cos uz dz du

To Jo Jo

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po cos ux du T

- sin ax ; Jo a? – 2a which result is the same as (177), Art. 108.

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Ex. 4. Let f (x) = | $ (y) e- dy; and let us suppose f (x) = f(-x); then from (145), (°°(y)e-audy = 46°4°°(4) e-* cos uz cos uz dydz du. (150)

Jo Ja The right-hand member of this equation is a triple integral, in which all the limits of integration are constant. Consequently, as we have demonstrated in Art. 99, the truth of the result is independent of the order in which the integrations are effected; and since by (6), Art. 82,

e-2 cos uz dz

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Section 1.-On Double, Triple, and Multiple Integration.

206.] Although hitherto in this treatise the element, whose integral is to be found, has been primarily an infinitesimal of the first order, yet it is evident, and it has been so stated in the first article, that such need not always be the case. Infinitesimals of higher orders may be subjects of integration equally as much as those of the first order. Several times indeed, but only incidentally, have such arisen in previous investigations; and subject to a particular condition whereby we have avoided a difficulty which is inherent in those of a more general character. Elementfunctions of an order higher than the first have been investigated in Art. 99, and it has therein been shewn how they may be the subjects of successive integration; and in the inquiry into definite integrals that theory has been frequently applied; as in Cauchy's method of evaluation in Art. 103; in the investigation of the properties of the Gamma-function, and in the application of those properties, and in Fourier's Theorem. Also again in Chapter V, an element of the nth order in terms of dx, involving an elementfunction of a single variable, has been the subject of n successive integrations.

Now in all three cases the limits of integration have been constants; and thus, as we have demonstrated in Art. 99, the order of integration has been indifferent. These circumstances have indeed removed the difficulties of the subject, but they have also restricted the problem within very narrow limits. It is necessary however to consider it in its utmost generality. Very many subsequent applications of the Calculus depend on multiple integration; and perspicuity is required in this subject more perhaps than in any other. I propose therefore to consider the theory generally in this chapter, and to apply it to geometrical and other problems in subsequent chapters.

207.] In the general problem of multiple integration the infinitesimal element is a product of factors one of which is a function of n variables independent of each other, and the others are differentials of these variables, each of whch enters in the first degree only. Thus if X1, Xg, Kg, ... Xm, are n independent variables, the infinitesimal element is of the form, F(xy, Kg, ... Xm) dx, dx, dxg ... dxm.

(1) This is evidently an infinitesimal of the nth order; and consequently n successive integrations must be effected on it as their subject-function, before we arrive at the finite quantity, which is the object of search. And if the problem is definite, these several integrations must be definite; and it is also necessary that the element-function should not become infinite or discontinuous for any value of a variable within the several ranges of integration. In the preceding parts of our treatise, wherever an element of the form (1) has been the subject of integration, the limits of integration have been constant; in the general problem however this will not be the case; for the limits will be, or may be, functions of all the variables which enter into the element-function of that integration. Now these are new circumstances requiring an extension of language and an extension of symbols.

Integration, as heretofore, means the summation of a series, and the limits of integration assign the first and last terms of the series. When an element of the form (1) is the subject of integration, and the finite function is sought of which (1) is the infinitesimal element, n summations must be effected, and in a prescribed order; let us take the order of integrations to be that in which the differentials stand in (1); so that the integration is first to be effected with respect to xz; let Xy, x, be the limits of this integration; let X,-x, be divided into n infinitesimal parts; and let &1, 62, ... En-1 be the values of x, corresponding to the points of partition. As X, is independent of the other variables, viz., Xg, Wgy...Xm, which enter into the element (1), so will &1, $, ... Én-1, which are particular values of x,, be also independent of these variables ; and consequently in the summation of the series found by giving X, successive values, X2, X3, ... Xn, and also dag, dxg, ... dxn, are to be considered constant. Thus the series which is first to be summed is of the form { F(X1, X2, ... Xn) ($1-X,)+F($138 g, ...In) ($-$1)+...

... + F (En-1, X2, ... Xm) (X,-&u-1)} dx, dxg ... dx»; (2)

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