order; being, in fact, quadrilaterals, each of whose sides is an infinitesimal, and which are therefore infinitesimals of the second order. 97.] Of the theorems of the preceding Article, that contained in equation (89) is the most useful for our present inquiry, although the others are important in some subsequent physical investigations. Now (89) may be expressed in the following form. (*n d.F(a, x) da de [**r' (a, x) dx = da dx. (92) Hence it appears that the differential of a definite integral with respect to a variable parameter involved in the element-function is the definite integral of the differential of the element-function with respect to that variable parameter. The two operations therefore of differentiation and of integration, effected as they are with respect to different variables, and thus independent of each other, may be interchanged without any alteration of the result. The process of differentiating a definite integral with respect to a variable parameter involved in its element-function having been thus established, the operation may be repeated on the function thus found: and repeated as often as the circumstances require; and as the order of these operations is indifferent, the final result will be expressed in the following form; The complete expression given in (91) may of course be subjected to repeated differentiations in the same manner as the part of it given in (92). But as we shall not require the general result, it is not necessary to insert it here. 98.] The following are examples of (92) and (93), in which from a known definite integral by means of differentiation with respect to a variable contained in its element-function a new definite integral is evaluated. Ex. 1. Let the known definite integral be Ex. 3. Since, if a is greater than b, by (27), Art. 67, therefore differentiating with respect to a, and reducing, (94) 2 a = + tan-1 a2-b2a+b cos x (a2 — b2) the latter result is also the a-differential of (46). - (a2 — b2) + therefore differentiating both members n-1 times with respect to a, we have taking successively the a- and the b-differential, we have If we differentiate this integral again with respect to a and b successively, and add the results, we have the same process may be repeated, and other integrals will be evaluated. 99.] As a definite integral may be considered a continuous function of a variable parameter involved in the element-function, and thus be the subject of differentiation with respect to this parameter; so may it also be the element-function of a definite integral, when the arbitrary parameter continuously varies: the element-function being finite and continuous for all employed values of the parameter as well as for those of its original subject-variable. Let u be the given definite integral; then u = F'(a, x) dx = = (X1−x ̧) F′(α, x ̧) + (X2—X1) F′(α, X1) + ..... Now let us suppose the parameter a to vary continuously from Q ао to a, and let the distance a,,-a, be divided into n infinitesimal parts, to the points of partition of which let a1, ag, ... a-1 correspond; then · + (an—a‚_1) F′ (an−1,Xo)} +(X2−X1){(α1—a。) F′(α, X1)+(α3—α1) F′(α1, X1) + ..... +(an—a,-1) F′ (an_1, X1)} + +(xn−Xn-1) { (α1— α) F′(α, X‚−1) + (α2 —α1) F′(α1, xn−1) + ... . . = (α1—α) { (X1—X0) F′ (α, Xo)+(X2 — X1) F′ (α, X1) + ... ...+(x-x1) F' (αo, Xn-1)} n +(α2−α1) { (X1−X0) F′(α1, Xo)+(X2— X1) F′(α1, X1) + ... +(a„−an−1) { (X1−xo) F′(αn−1,Xo) + (X2−X1) F′ (an_1, X1) + . . . +(xn−Xn-1) F′ (an_1, Xn-1)} (101) = (a−q) [** 1′ (a, x') dx + (az − a ) (*ˆ x' (a,, x) dx + ... - = [** dx [** ¥′(a, x) da. απ +(a,—an-1) F′(an_1, x)} dx (102) Thus the a-integral of the x-integral of the element-function is equal to the x-integral of the a-integral; and thus the order of these two processes, effected as they are with respect to variables independent of each other, may be interchanged without alteration of the result. This theorem is called the inversion of the order of integrations; and the members of (102) are called double integrals. any In the preceding investigation it is assumed that a and all its values are independent of x and all its values. The inquiry into the properties of double integrals will be extended to the cases where this restriction is not made; but the preceding is sufficient for our present purpose. The principle of single integration of a definite integral with respect to a parameter contained in its element-function having been thus established, and the order of the operations having been shewn to be indifferent, the process may be repeated, subject of course to the same conditions. So that we have to r successive integrations, an an an = [** dx fˆˆ da f** da ........ “* x'(a, x) da. (103) Integrals of this kind are called multiple integrals, of which double, triple integrals are particular forms, according as two, three,... integration-processes are involved. The general theory is of great importance in subsequent investigations, and will be a subject of inquiry hereafter: but the preceding is sufficient at present. 100.] The following are examples of the process. [defe-"dade = log. (105) dx = dx |