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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., X. Xn-1 X, X,

7 F(X1, X2, ... X ) dx dx 2 ... dx »; (46) and let us suppose the integral to include all values of the variables for which the function f (x1, X2, ... Xo) is negative; so that X, and x, are functions of all the variables except X1; X, and x, are functions of all the variables except X, and Xı; and so on : X.-, and x.n-1 being functions of X, 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

En fan-1 11. I=

*10 ($1, $2, ... Ex) dx d2 ... dfn; (47)

also suppose the equation, which determines the limits, to become 0({ $2, ... En); so that the integral includes all values of $1, $2, ... É for which $($1$, ... &n) is negative; then the limits must be determined as follows;

Observing the order of the successive integrations as indicated by the order in which the differentials are arranged, E, and E, are functions of all the variables except &, and are determined by putting 0 = 0, for thus will be included all values of &, for which o is negative. By this process the new element-function will involve $2,£3, ... En; and the limits of integration must include all values of the variables for which $i 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 $v, as found from $ = 0, are equal ; that is, by the simultaneous equations p = 0, and () = 0. From these two equations therefore if we eliminate &, we shall have a resulting equation of the form 01 ($2, 63, ... &n) = 0, from which E2, E, are to be determined, because they are the values of $, which satisfy 0 = 0; and thus we shall include all values of &, for which the expression $ı is negative. Similarly by eliminating $2 between the equations $i =0, and (ht) = 0, the limiting values of &z will be determined. And the limits of all the other variables will be determined by similar processes.

Hence in recapitulation; the limits of & are determined by solving for & the equation =0; those of E, by solving for & the equation which results from the elimination of fi between $= 0, and

); those of &z by solving for $g the equation which results from the elimination of &, and {z between $ = 0, (de) = 0, and ( 0 ) = 0; and so for the others.

If the equations $ = 0,01 = 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

1=(, z) dz dy dx,

all values of the variables are to be included for which

m2 2 22
O2 + 72 + 2 -1 = f(x, y, z)

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

= + 2, say. The y-integration comes next; and its limits are given by the elimination of z by means of f(«, y, z) = 0, and (.) = 0; whence we have

mean

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= + Y, say; and the x-integration comes last; and its limits are given by the elimination of z and y by means of f (x, y, z) = 0, = 0 = 0; whence we have

2 = + a. so that

ra ry z

| F(x, y, z) dz dy dx.

J-a J-y -z 215.] Examples of transformation of definite multiple integrals.

Ex. 1. The double integral //(x2+y2) dy dx extends to all points within the area of the circle x2 + y2 = a’; it is required to express it in terms of polar coordinates. If y=(as – 22),

1= S(x2+yo) dy dx = 1** (*-* dr do. (48)

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Ex. 2. Transform into its equivalent in terms of polar coordinates the double integral

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(53)

= f(m+n) B(m, n), as appears from (50), and from (271), Art. 126. Hereby the fundamental theorem connecting the Beta- and the Gamma-functions is established.

Again, take (50), and let m = 7; then

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(54)

:: () = mid; the same result as 280, Art. 129.

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