2. Fa, b2 ... Prdens And proceeding by processes similar to the above, dx, = 2. + a, b, ... r, and therefore by substitution Anzota, ß2 ... Pn de dx, dx , ... dx, = (-)" 2 --- ** df, df2 ... den: (28) 2. + a, b, ... rnus Equation (19) is evidently only a particular case of (28); that viz., in which a, = b2 = cg = ... = rn = 1, and all the other partial derived functions vanish. Hence we have the following theorem ; (29) ana), Cenas). Cara), Cena).... (30) The sign of the result being ambiguous, because it depends on the order of integration in the transformed integral, and that order is obviously arbitrary*. 213.] The following are examples illustrative of the preceding theorem. * A full discussion of the theory of the transformation of multiple integrals will be found in a paper on the subject by M. Catalan, in the Mémoires couronnés par l'Académie de Bruxelles, Tome XIV, 1839, 1840. Ex. 1. Take the double integral/ | F(x, y) dy dx ; and (1) let the equations of transformation be X = 6 cos a-n sin a, y=&sin a + n cos a. dy = sin a df+cos a dn; (31) (2) Let the equations of transformation be x = r cos 0, y = r sin 0. dy = sin 0 dr +r cos 0 do;S (32) These cases are obviously those of transformation from a rectangular system of axes, to another rectangular system and to a polar system respectively. (3) More generally, if the equations of transformation are given in the form, x = fi(,n), y = fg (8,); then (33) de = ( de las) of+ a ) dan 1 dy = (alfa) s + cenadan ; } :: de dy = + {care o f a real )} dę dn. (34) Ex. 2. Let the integral be the triple integral Ex. 5. To transform into its equivalent dx dy dz, when x=lr, y = mr, 2 = nr; l, m, and n being subject to the condition 12+ m2 + m2 = 1. By a process similar to those above, we have the following results: 2 ga dr dm dn gol dr dn di pe dr dl dm (45) m 214.] If the original integral is definite, the transformed one will also be definite; for the latter is to be equivalent to the former in all respects, and consequently the values of the variables in both integrals must extend over the same district of |