504 -atD d sin u cos v + COS U ; Now = a Dt["de sin 0 eat D cose ; but as the function Q is arbitrary, we may write o instead of aDo, so that (ctD - 6-4D) P(x, y, z) = {* do sin cat D cos® + Q (x, y, z). Now by Ex. (4, d) 27 5" do sin 8 ? D cos e alla sin u sin v + = 6"/9* du dv sin u e su) therefore -at) Q (x, y, z) 1," ),** du dv sin u.t.4(v + at sin u sin v, y + at sin u cos v, % + ai cos u). d (a D) (catD - 6-46D), - 6-4), we find dt 10" ,** du dv sin u.t. p(x+ at sin u sin v, y+ at sin u cos v, 3+ at cos u) 4 을 So" 16** lu dv sin u.t.4(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. at D - € Also as at D 6 -(D te -1 4πυ – + (c) The equation for determining the vibratory motion of a thin elastic lamina is d? Hence putting bt dir? for a”, we Therefore as F(x) is an arbitrary function, and as da may write F(x) for 6 F(x), we have sdy cos (– yj) {v – 2y (64)!} + sdt sdy cos (5 – y*) F{x – 2y (64)*}. Poisson, Ib. 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 Tydx and sudy, to compare certain transcendents of considerable interest. Since Firdy + Jydx = xy + const., if a symmetrical equation subsist between x and y, so that 2 is the same function of y that y is of 2, or that when p(y), y = 0 (2'); it follows that SP(x) dx + 50 (y) dy = xy + const. This is true whatever be the nature of p, independently of the integrability of the functions. (7) Thus if x be the abscissa of a hyperbola, the major axis of which is unity, the corresponding arc is represented by (e being the excentricity). If y be another abscissa connected with the former by the equation e x? ey = - (:') or e“ v? ye - e' (x2 + y') + 1 = 0, which is symmetrical with respect to x and y, it follows that ey-1 y - 1 Therefore ers (7) . e' yi - 1)? =exy + const. y - 1 Mr Fox Talbot * has extended to any number of variables the principle made use of by Fagnani in the case of two, and he has arrived at the following Theorem. If there be n variables x, y, x, &c. connected by (n − 1) symmetrical equations, so that they are all similar functions of each other, then if we shall have SP(x)dx + 5°(y)dy + 5°(x)dx + &c. = xyz &c. + const. • Phil. Trans. 1836 and 1837. The same theorem in a somewhat different form had been previously given by Hill in Crelle's Journal, xi. p. 193. It is only a case of the more general one in which the continued product xyz, &c. is replaced by any symmetrical function of those quantities. Let the variables be three in number, and let the symmetrical conditions be x + y + z = xy + y + 2x + 3 X Y Z + 1 = 0. Then since (y+) dx + (x + x) dy + (x + y) dx = d (xy + yz + xx), and since by the conditions just given 1 3.0 +2° y + z = x (x - 1) we shall have 3.8 + rez 1 - 3y + y" 32 + x dy + dz x (x - 1) y (y - 1) (x - 1) EXY + yx + 2x + C, a result easily verified. dir + If the two conditions be x + y + x = 0, (x+ – 1) (y – 1) (x2 - 1) + 1 = 0, we shall find that Hence by the theorem 1+y+-Y' 2) 1++ 1-? since x + y + z = 0 by the first condition. (2) The principle of symmetry, of which these examples afford an illustration, is of the greatest importance in the |