represents a closed surface; then, if p= q = r = 1, the given definite integral expresses the volume of the octant of that surface, and if v = the whole volume,

281.] Soon after the publication of the preceding theorem by Dirichlet, M. Liouville* gave an extension of it to those cases in which the element-function involves an arbitrary function of that particular combination of the variables which is given in the following form,

I =

· f (x + y + z + ...) x2-1 yî-1 z1-1 ... dz dy dx; (46)

where the range of integration includes all positive values of the variables given by the inequality

x + y + z+... <h,

where h is a positive constant.

Let us first take the case of two variables only; then

(47)

(48)

0

Let this integral be transformed, and let x = uv, y = u (1—v) ; then, as in (55), Art. 215, dy dx = u du dv; and taking the limits of the new variables so that the same range may be included,

As the limits of both integrations are constant, the order in which the integrations are effected is indifferent; and consequently

so that the double integral given in (48) is reduced to the product

of a certain combination of the Gamma-function and of a single

definite integral.

* Liouville's Journal, Tome IV, p. 230.

Next let us suppose three variables to enter into the integral (46); so that

To reduce this, let a transformation be made of two variables similar to the preceding; and let y = vw, z = v(1-w); then, dy dz= v dv dw; and

I=

=

"h h-x

r (q) r (r) [* [^ ̄*ƒ (x + v) x2−1 y1+r−1 dv dx ;

r (q+r) Jo Jo

:

in which the definite integral involves only two variables and transforming again as in the former cases we shall have

This process may evidently be extended to any number of variables; and we shall have ultimately

y?−1 zr−1

·ƒ (x + y + z + . . . ) x2-1 yî—1 z1-1 ... dz dy dx

=

..

r (p) r (q) r (r') ... [ * ƒ (u) w+e+r+.....-1 du ; (53)

r(p+q+r+...) Jo

where the limits of integration are given by the inequality

so that by this theorem the multiple integral is reduced to a single integral.

If ƒ (u) = 1, we have the result already given in (23).

f

282.] This theorem also admits of extension as to the inequality which determines the limits, similar to that of Art. 280 from Art. 279. In this case

where the range includes all positive values of the variables which are given by the inequality

(+++

<h.

(55)

Let us take the case of three variables, so that we have the integral

and let us make the substitutions which are given in (26) and (28); so that the range includes all values of the new variables έ, 7, Ŝ which are given by the inequality

If f(u) = 1, and h = 1, we have the result already given in (29); so that this theorem includes all those previously given. The following are examples in which it is applied.

of f f ( x + x2 + y2
G dy dx, where

Determine the value of

1+

the limits are given by the inequality x2+ y2 <1.

Here we have only two variables; and p=q=1; a=ß = 2,

a = b = 1; h = 1; so that if y = (1 − x2)3,

0

+

4

0

π(π-2)

=

du

(61)

where

Ex. 2. Determine the value of

SSS

8

dz dy dx
(1 − x2 — y2 — z2) §

the limits are given by the inequality x2 + y2+z2<1. In this case the range includes all points lying within the surface of a sphere whose radius = 1. If z = (1 − x2 —y2)3, y — (1—x2)1,

where the limits are given by the inequality x2 + y2+z2 < 1.

Taking the same notation as in the preceding example,

SECTION 2.-Dirichlet's Method of Reduction by means of a Factor of Discontinuity. The Application of Fourier's Integral.

283.] An examination into the theory of multiple integrals at once shews that generally the difficulty of evaluating those whose limits are given by an inequality which involves the variables is much greater than that of evaluating those of which the limits are constant. In the latter case, as we have demonstrated in Art. 99, the order of the integrations may be changed without any change in the value of the result, and in very many cases we are able to simplify the integral by means of properties of the Gamma-function, and of other allied integrals, which have been proved heretofore. In the former case however the order of integration is prescribed, and cannot be changed without (in many cases) considerable difficulty and consequent risk of error; this fact is apparent from the difficulty of assigning the limits of integration when the integral is transformed by a change of variable. Now the knowledge of this circumstance appears to have suggested to L. Dirichlet the process of so operating on the infinitesimal element-function of a multiple integral, of which the limits are assigned by an inequality, that the limits may be constant; and indeed as he has shewn, and as it is convenient to take them, that the limits of all the several integrations may be and 0. To effect this, he introduces a factor in the form ∞ of a definite integral, into the element-function; this factor being a discontinuous function which 1 for all values of the variables within the range of integration, and = 0 for all values of the variables beyond that range. Consequently when this factor has been introduced, we may enlarge the range of integration to any extent; and may indeed include all values from to 0; or from ∞ to -∞. We have already had similar cases in which the range has been so enlarged; see Art. 197. The mode of applying this principle is as follows; Let the given definite integral contain n variables, and be of the form

I

=SSS ::

... F(x, yz,...)...dz dy dx;

(64)

and let us suppose the integral to include all positive values of the variables within limits assigned by the inequality

Now suppose that we have a single definite integral of the

form

[*ƒ (t, k) dt, containing the undetermined constant k; and

that this integral = 1 for all values of k less than 1, and = 0 for all values of k greater than 1; then if we replace k by x+y+z+..., this integral will be equal to 1 or 0, according as we are considering values of the variables within or beyond the range assigned by the inequality (65). Consequently if we introduce within the integration-symbol in (64) the factor

[f(t,

f(t, x+y
,x + y + z + ...)dt,

(66)

we may enlarge the limits of all the other integrations to ∞ and O without changing the value of the integral. Thus we have

F(x,y,z,...)f(t,x+y+z+...)...dz dy dx dt. (67)

...

The case in which k = x+y+z+ = 1 will only give one element of the definite integral, when the limits are extended; and consequently, assuming that the introduced factor is finite when k = 1, that element must be neglected in the definite integral; and it is necessary only to take account of the elements when k is greater than and less than 1. This remark is important; and is applicable to all the subsequent cases in the present section where the limits of integration are given by an equality, and the limit of the value of this inequality is expressed by a discontinuous factor.

Now a factor satisfying the preceding conditions is called a factor of discontinuity. A quantity possessing these qualities has been investigated in Ex. 4, Art. 100; in which it is shewn "sin mtcoskt 1

that

2

π

t

dt=1, for all values of k less than m; ='

2

sin t cos kt

t