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287.] An important theorem may be derived from (88). Let a, b, c, and u be replaced by Oa, 06, 0c, ou respectively; and to shorten the formulæ, let us take only two variables: then, from (86) and (88), the limits being given by u < x+y<d, 122179-1 e-(az +by)® f(x+y) dy dx = the real part of P 9 /P(t) dt / _ cos Out du

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Let each side of this equation be multiplied by 9P+9-12- do, and let the O-integral of each side be taken for the limits oc and 0; then

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therefore for the limits assigned by the inequality, < x+y<i,

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- 29–179-1... dy dx. (99) {w+lxa + myß +... ...“ The process of reducing this to the form (97) is exactly similar to that of Art. 280; and it is unnecessary to repeat it. The following example will illustrate it. Let it be required to determine the value of I dz dy dx .

(100) J (w + x2 + y2 +22) where the limits of integration are given by the inequality,

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(101)

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u* du 2 Ju {(au? +w) (62 u2 +w) (c^u2 +w)}} and thus the triple integral is reduced to a single integral.

289.] Thus far, when the infinitesimal element has contained an arbitrary function, the subject of that function has been the expression by which the inequality assigning the limits of integration has been determined. Let us consider the following

integral in which this is not the case; and here, as before, I will take an integral involving only three variables ; viz.

1 = // /F(x+y+z) dz dy dx, where the limits are assigned by the inequality,

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so that the integral includes all values of the variables which lie. within the surface of an ellipsoid. Let

1. x2 2 22

O2 + ž2 + 2 = k; x+y+z = 8. (103)

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But by (362), Art. 140,

56*su)vide = 0 (5)*(1-3); and similar values are true for the y- and the z-integrals; so that substituting these, and replacing a2 + b3 + c3 by me, we have SIF(x+y+z) dz dy dx =

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whereby the triple integral is expressed in terms of a double integral.

If the integral contains n variables, the limits being given by the inequality,

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and thus the evaluation of the multiple integral depends on that of the double integral.

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