r(p)r(qr(r)r(m-p-9-r). (14) r(m) From (13) a more general theorem may be deduced by replacing ax by (9)", By by (3)", yz by (3)'; but it is unnecessary to express it at length. If in the general theorem (12) there is only one variable x, po x-1e-ax dx r(p) po tm-le-kldt (15) Jo (k + ax)" r(m) J. (a + at) In this equation let k=a=1; and in the right-hand member let t be replaced by x; then (16) pos XP-le-* dx pogum-le-- dx Jo (1 + ax)* po mp-1 dx_r(p) m-p-1e-dt (17) r(in) a result which has already been found in Art. 125. 279.] Let us now consider these cases in which a reduction of a multiple integral can be made by means of the Gamma-function, and in which the limits of integration are given by an equation of condition. The integral which I shall take is the following of n variables. I=\|...x-179–121–1...(a—x-Y-2–...).–1.dzdy dx ; (18) the limits of integration of which include all positive values of the variables satisfying the inequality x+y+z+ ..... <a. (19) Let us however confine ourselves to three variables ; for we thereby fix our thoughts, and do not restrict the nature of the process of reduction which is the same, whatever is the number of variables. In this case the equation of limits gives the surface which bounds all those points in space which are included in the integral; and in this special case that surface is the plane whose equation is x +y +z = a. Hence para-x pa-s- , 29-149-12"-1(a-2-y-z).-1 dz dy dx. (20) Jo Jo Jo Now by (270), Art. 126, m+-1 T(m) r(n) | U"?(C-U)"-1 du = (21) r(m+n) so that applying this theorem to the successive integrations in (20), we have 1=1*7*700-19,9–1(a—~—}+c-15 (v)r (8) dy do r(r+8) r(q+r+8) r(q+r+8) (P+9+8+8) (22) r(p+9+8+8) which result completely determines the value of the integral given in (20). If s = 1, XP-249-1 2-1 dz dy dx = r(p)r (9) r(r) ?+; (23) JOO JO r(P+9+r+1) all positive values of the variables being included which lie within the plane whose equation is x+y+z = a. 280.] This theorem is capable of extension, and in its extended form supplies the solution of many important problems. For we can by a similar process determine the value of XP-149-12-1 dz dy dx ; (24) when the limits of integration include all positive values of the variables given by the inequality where p, q, r, a, b, c, a, b, y are all positive quantities. so that the inequality (25), which assigns the limits, becomes $+n+5 = 1; (27) This useful theorem was first given by Lejeune Dirichlet *, having been deduced by him from a general process which will be described in section 2 of the present chapter; the following are examples in which it is applied. Ex. 1. Let there be two variables, and let a = B = 2; then the equation which assigns the limits is (*)* + ( )* = 1; and * Comptes Rendus, Tome VIII, p. 159 ; 1839. PRICE, VOL. II. consequently the range includes all values of the variables which correspond to points within the first quadrant of an ellipse; and if y = (22 –x2), (30) (31) Jo Jo And as the left-hand member evidently expresses the area of a quadrant of an ellipse, the area of the ellipse = Tab. (32) and consequently for the whole ellipse, [L*(**+y?) dy dx = mat (a2 +6%). (37) All these expressions will be of considerable use in the sequel. Ex. 2. Let three variables be involved in the integral (29), and let a = B = y = 2, so that the equation which assigns the limits er et + = 1; (38) is and consequently the range includes all values of the variables which correspond to points within the first octant of the ellipsoid. Let p =q=r=l; then fayz παbe Idz dy dx = 6 (40) Jo Jo Jo As the left-hand member of this equation expresses the volume of the octant of the ellipsoid, the whole volume of the ellipsoid (42) If p= 3,q=r= 1, then Pa ry z a al bc 30 And for the whole ellipsoid 47a3 bc (43) . Similar equivalents are of course true for the other variables. Ex. 3. Let the definite integral (29) contain three variables ; and let a =B=y= 4; so that the equation which assigns the limits of integration is (44) then, if p =q=r=1, 1 in (29) expresses the volume of the octant of the surface whose equation is (44); and if v is the whole volume, V = 6.25 TL which, as shewn by (57), Art. 162, may be expressed in terms of the arc of a lemniscata. Ex. 4. Generally if the definite integral contains three variables X, Y, 2, and à, B, y are even numbers so that the equation |