Now if u be a function which, with its differentials up to the (1)th, vanishes at the limits, we have, on integrating r times by parts between the limits, we have, putting ≈ cosa and dx = - sin xdx, (r) /*daf\r (cos).(sin®)*=1.3.5...(2r−1)*dæf(cos®).cosr. (26) I shall conclude this Chapter on Definite Integrals with some examples of their application to the solution of partial differential equations. This mode of expressing the integrals of such equations was introduced by Laplace*, and has been much employed by later writers, particularly Poissont, Fourier‡, Cauchy, and Brisson || . It is particularly applicable to linear equations of orders higher than the first with constant coefficients, and it is useful because the solutions are put into a shape which facilitates the determination of the arbitrary functions. The principle of the method is to transform an explicit function not expressible in finite terms into an ordinary function involved under a Mémoires de l'Académie, 1779. + Mémoires de l'Institut. 1818, and Journal Polytech. Cah. x11. Théorie de la Chaleur. definite integral; but the mode of transformation must be determined in each particular case by the nature of the function to be transformed, as will be seen in the following examples. (see Chap. VI. Sect. 1. Ex. 4. of the Integ. Calc.), and our object is to transform the operative function into one involving Therefore, putting a d dx Now and to dwe (w—b)2 = π3. t for b, and multiplying the two sides of the integral by the two sides of this equation, therefore π3 x = [_+ * dwe ̄w2 ƒ (x + 2 wat3). This transformation is due to Laplace, Jour. Polytech. at D V = ε F(x, y, z) + € ̄at D ƒ (x, y, z); which may also be put under the form at D 2v = atD − e−atD) p (x, y, x) + (ea¿D + €−a‹D) ¥ (x, y, z), if ø + y = F, $ − ¥ = ƒ. π D (x, y, ≈) = ƒ de sin at coset(x, y, z). at (d sin u sin v + dx 2π (eatDe-atD) (x, y, z) = π 2π ST £2 du dv sinu.t. (x + at sin u sin v, y + at sin u cos v, z+at cos u). This transformation is given by Poisson, Mémoires de l'Institut, 1818. (c) The equation for determining the vibratory motion of a thin elastic lamina is = COS (bt the integral of which is dae) f (x) + f** dy e-2ay cos y2 πcos (a2 + d2 Also sin (bt)-bfdt cos (bt). = dx2 Therefore as F(x) is an arbitrary function, and as we + fdt fdy cos ( − y2) F {x - 2y (bt)}}. Poisson, Ib. CHAPTER XII. COMPARISON OF TRANSCENDENTS. THE integration of differential expressions frequently leads to forms which are not expressible by any finite combination of algebraic, circular, and logarithmic functions. Such integrals are called transcendents, and the study of their properties becomes of importance as affording the means of classifying and arranging them so as to reduce them to the smallest number of independent functions. The class of transcendents which has been most studied consists of those called elliptic, from their being in certain cases capable of representation by elliptic arcs. They thus appear to be functions little more complicated than those which are represented by circular arcs, and to be naturally pointed out as the next subject of investigation. The properties of these functions which have been discovered, relating chiefly to sums and differences of connected transcendents are very numerous; but in the following pages I shall confine myself to elementary illustrations of some of the principal theorems, making use chiefly of those examples which admit of a geometrical interpretation. Fagnani has availed himself of the relation which subsists between the integrals to Since compare fyda and fady, certain transcendents of considerable interest. fxdy + fydx = xy + const., if a symmetrical equation subsist between and y, so that a is the same function of y that y is of a, or that when |