values. Thus the principle on which the limits of the new integral are to be assigned is the same in all cases, although the particular values will depend on the peculiar circumstances of each case. The limits of the original integral, and the equations of relation, will be found sufficient for the determination of the limits of the transformed integral. The following case frequently occurs, and illustrates the principle. Let the definite integral, which is the subject of transformation, be of the form (8), viz., 2 and let us suppose the integral to include all values of the variables for which the function f(x1, x2, ... x) is negative; so that X, and x, are functions of all the variables except a1; X2 and x2 are functions of all the variables except x and x1; and so on: X, and x, being functions of a only; and X, and x, being constants. Now suppose 1, 2, ... En to be new variables, connected with the former variables by equations of relation; and let the new integral be §2, also suppose the equation, which determines the limits, to become ({1, 2, ... n); so that the integral includes all values of 1, 2, ... E for which (E1, E,... En) is negative; then the limits must be determined as follows; $(§1, Observing the order of the successive integrations as indicated by the order in which the differentials are arranged, E1 and E1 are functions of all the variables except έ1, and are determined by putting = 0, for thus will be included all values of έ, for o which is negative. By this process the new element-function will involve 2, 3,... έ„; and the limits of integration must include all values of the variables for which έ, is not impossible. To determine this condition, let it be remembered that when, by the variation of certain quantities, the roots of an equation having been real become impossible, in the course of their change they must have been equal; the limits therefore of the new integral will be given by the conditions for which two at least of the values of 1, as found from = 0, are equal; that is, by the From these two simultaneous equations 4 = 0, and ( d ) = 0. d. equations therefore if we eliminate 1, we shall have a resulting equation of the form 41 (2, 3,... E)= 0, from which E, E are $3, Śa) to be determined, because they are the values of έ, which satisfy $1 = 0; and thus we shall include all values of § for which the expression is negative. Similarly by eliminating §, between the equations 41 = 0, and (a O, the limiting values of § will (dipy) = be determined. And the limits of all the other variables will be determined by similar processes. Hence in recapitulation; the limits of 1 are determined by solving for the equation = 0; those of 2 by solving for ₤2 the equation which results from the elimination of έ, between аф 4 = 0, and (do ) = 0; those of § by solving for §, the equation αξι which results from the elimination of έ, and аф between = 0, do) = 0, and (d) = 0; and so for the others. dé1 If the equations = 0, 1 = 0, , give only two values of 1, 2,... for the limits, then the case is free from difficulty, and the new definite integral assumes the form in which it is written above but if any of these equations has more than two roots, and if there is nothing in the conditions of the problem which excludes them, we must resolve I into a series of integrals according to the roots, and the limits of the several integrals will always be given by the equations found as above*. As an example, let us suppose a triple definite integral to include all variables which refer to points lying within the surface of a given ellipsoid, and to exclude all others. Then if the triple integral is I = F(x, y, z) dz dy dx, all values of the variables are to be included for which is negative. Now if the integrations are to be effected in the order in which the differentials are placed, the z-integration * See a Memoir by M. Ostrogradsky on the Calculus of Variations in the Memoirs of the Imperial Academy of St. Petersbourg, Vol. I, 1838, p. 46. The Memoir has been reprinted in Crelle's Mathematical Journal, Vol. XV. P. 332. comes first, and its limits are given by f(x, y, z) = 0; whence we have c{1 x2 y2 ) 1 The y-integration comes next; and its limits are given by the elimination of z by means of f(x, y, z) = 0, and dz and (df) = 0; and the x-integration comes last; and its limits are given by the elimination of z and y by means of ƒ (x, y, z) = 0, (d) = 0, (d) = 0; whence we have 215.] Examples of transformation of definite multiple integrals. Ex. 1. The double integral [f(x2+ y2) dy dæ extends to all points within the area of the circle x2+y2 = a2; it is required to express it in terms of polar coordinates. If y = (a2 — x2)3, x =ƒa [*(x2 + y2) dy dx = [** ['*r3 dr do. I -Y 0 (48) Ex. 2. Transform into its equivalent in terms of polar coordinates the double integral Since xr cos 0, y = rsine, therefore by (32), dy dx = rdr do; also the given limits shew that the district of the integral is bounded by the equation a2 y2+ b2 x2 — a2 b2 = 0; and this expressed in polar coordinates is r2 { a2 (sin0)2 + b2 (cos 0)2 } — a2 b2=0; ab Ex. 3. If z2 a2x2-y2, y2 = a2-x2; then in terms of polar Ex. 4. In the definition of the Gamma-function given in (247), Art. 122, let x be replaced by x2; then In this definite integral let x = r cos 0, y = r sin ; then as the limits of x and y are such that all positive values of x and y, that is, the values of x and y corresponding to all points in the first quadrant of the plane (x, y), are included in the integral, the limits of r and must be co-extensive; hence r (m) r(n) = 4 // e-22m+2n-1 (sin 0)2n-1 (cos 0)2m-1 dr d0 (52) S = 2 [" e-r2 42m+2n−1 dr × 2 [ " (sin 0)2n−1 (cos 0)2m−1 d0 = г(m+n) B(m, n), (53) as appears from (50), and from (271), Art. 126. Hereby the fundamental theorem connecting the Beta- and the Gamma-functions is established. Ex. 5. Double integrals of certain forms are frequently simplified by the following formulæ ; viz. x = u(1—v), y=uv. From these values we have = da (1-v) du-u dv, dy = v du + u dv. (55) .. dy dxudu dv. The following is the geometrical interpretation of this substitution. In fig. 44, let p be (x, y); through P draw PN parallel to, and PS making an angle of 45° with, the axis of x, and join NS; then osu, and tan osNv: for Hence, if the limits of integration of the double integral are given, it is easy to assign the limits of the transformed integral. Thus, taking the definition of the Gamma-function given in (247), Art. 122, we have integral [[a"y"f(x+y) dy da By means of this substitution the is changed into ffum+*+1 (1—v)TM v* ƒ (u) dudo. |