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and the sum of this series, if F (X2, X, ... ~) dx, is the xn-partialdifferential of f (X1, Xg, ... Xx), is {f(X, X, X z, ... Xn)-f(xy, X2, Xg, ... 2)} dx , dx, ... dxmı (3)

In the general case X, and x, will be functions of Xg, Kg, ... Xin, that is, of all the variables in the original function except X7

This process is called the integration of the element with respect to x, or the x,-integration of the given element; X, and x, being limits of the x,-integration; and by it the element takes the form, f(xg, X3, ... Xn) dx, dxg ... dx m.

(4) This is again to be treated with regard to Xq, as the element given in (1) has been treated with regard to Xi; if X, X, are the limits of this second integration, they will generally be functions of all the variables except X, and Xg, and the result will be an element of the form, F(X3, X4, ... Xn) ddx 4 ... dx .

(5) This process is called the x,-integration of the element given in (4), and X, Xy, are called the limits of the x-integration.

The process is to be repeated for all the successive variables in the order in which the differentials are placed in (1); and if X, and s, are the limits of the xr-integration, which is the last, these will be constants, and the final integral will be of the form, F(X.) – F(xn);

(6) and will of course be independent of all the variables involved in the original element. This is the quantity which is the object of search.

208.] The following is the system of symbols which I shall adopt. Let the integral be generally expressed by

|| ..... /F(Xy, tz, ... q)dx, dx, ... dxn-, dxni (7) the order in which the differentials are placed being that in which the successive integrations are to be effected. Thus the xn-integration is the first, and the x,-integration is the last. If the limits of integration are given, so that the integral is definite, then it will be expressed in the following form ps, (x,1 X , X

F(X1, Xg, ... XndX, dx ... dx, : (8) Jx, Jxq Ixq Ix where X, and x, are, or may be, functions of all the variables except Xy; X, and x, are, or may be, functions of all the variables except &, and Xg: and so on; and lastly Xm, x, are constants. PRICE, VOL. II.


It will be observed that the order of the symbols of integration in (8) is the reverse of that of the differentials. The reason is, that the first definite integral becomes the subject of, and is thus included within the symbols of, the second integration; and so on. Thus dx, denotes an operation which is performed sub

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sequently to that denoted by / dx; and the quantity which is the result of the latter becomes the subject of the former, and is consequently properly included within its symbol.

Another arrangement of the symbols is the following, which plainly expresses the order of the successive integrations ;

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and as this system is sometimes more convenient than (8), it will be employed whenever that is the case.

Integrals of the form (7) are called multiple integrals; and those of the form (8) in which the limits are given are called definite multiple integrals. When n integrations are to be effected, they are called multiple integrals of the nth order. Particular forms of these are double integrals, triple integrals, ... according as two, three, or more integrations are to be effected. .

209.] For the truth of these theorems on multiple integrals it is necessary, as it has been before observed, that the elementfunction should not become infinite or discontinuous for any value of the variable within the ranges of integration. Subject to this condition the integrations may be effected in that order which is most convenient. In a definite multiple integral, when the limits are constant, the order of integration is indifferent, as we have demonstrated from first principles in Art.99; but if the limits are functions of the variables, the order cannot be changed without a change of the limits; and this is frequently a work of considerable difficulty ; because when a multiple integral is definite, it is required to find the sum of the values which the element has for all values of the variables within a certain district; now this district is assigned by the limits; and although it may theoretically be possible to determine the boundary of this district in many ways, yet practically one particular mode may be more convenient than another, or may be the only possible mode.

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Thus for example, suppose that it is required to find the value

dy dx for all points of the plane area contained within a cycloid and its base, the highest point being the origin, and the perpendicular to the base being the x-axis; then if we denote a versin-1 +(2ax— x2)# by Y; S | dy dx = '"* Lody de

= 2 *{a versin - + (2ax — x2}} dx

= 37a”; which is the required result. But the order of integration cannot be changed, because the limits of the x-integration cannot be determined from the equation of the cycloid; it is a transcendental equation, and x cannot be expressed explicitly in terms of y by means of it.

Theoretically however the order of integration is arbitrary, although the equation which fix the boundaries of the district through which the integration extends, may constrain us to take a particular order. Many illustrations of this remark will occur in the following chapters; wherein also will be shewn the convenience of one order in preference to another.

210.] The following are examples of definite multiple integrals.

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do 2 Ex. 4. Find the value of 23 dr do through the area bounded by the circle whose equation is (1) r = a; (2) r = 2 a cose.

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As the sequel will contain many examples of multiple integration, it is unnecessary to give others in this place.

SECTION 2.—On the Transformation of Multiple Integrals. 211.] The evaluation of multiple integrals may frequently be greatly facilitated by a change of variables. The process is called Transformation of Multiple Integrals ; and I propose in the present section to consider it in its most general form, and also to give various examples of it.

Let us take the integrals given in (7) and (8) to be the types of multiple integrals of the nth order; and, for the present omitting the limits, let

I= || | ...... | F(X1, X2, Xg, ... Un)dx, dx, ... dx (10) Let the new variables, in terms of which the original variables are to be expressed, be $u $2, ... Śm; and let us in the first place suppose the equations of relation to be given in the following explicit forms; viz.

X1 = fi ($1, 82, ...... $m), )
, = f, 1, 2, ...... 4),

(11) . . . . . . .

X n = fm ($i, $, ...... $m); J Let az, B., be the partial derived functions of x, with regard to &1, 62, ... Em respectively; and let Ag, B2, ... Pg be the similar derived functions of xz; and so on; then

dxı = a; dfi+B, d'fz + ...... tpadne 7
dx, = 0, d&i + B, d&2 + ...... +p, dere


(12) · · · · · · · · · · · · ·

dx, = a, d&,+B, d$2+...... +Pnden. J In article (207) I have explained how Xg, kg, ... X, remain con


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stant during the variation of X, that is, during the x,-integration; consequently in the calculation of the quantity which is to be substituted for dxı, it must be consistent with the conditions, dx, = dkg = ... = dxn = 0. Hence we have

dx = a;d&i+B, d&g + ...... tpid Sne
0 = a, d&i+Bzděz + ...... +P2d&n,

(13) · · · · · · · · · · · ..

0 = and $i+Bnd2+ ...... +puden; J whence, employing the symbols of determinants, dx, 2. B273 ... Pm = df, 3. † a BzY3 ..Pni 2. +9 32 73 ..... Pre de :

(14) 1. † B2 V3 ...... Pm the numerator of the right-hand member of which is the determinant of the system in the right-hand member of (12); and the denominator is the first minor determinant of the same system.

Again, in calculating dxg, dx, = dxz = dx, = ... = dxn = 0; and by reason of (14), dęı = 0 when dx, = 0; so that we have

dx, = B,df2+72 děz + ...... +p, dene
0 = B, df2+73 d&3 + ...... +P3 d&n,

(15) 0 = Bndfz+yndf3+ ...... + Prdfn;J whence, employing the symbols of determinants,

dx, 2. Ey3 ... Pn = , 2. B2 V3 ... Pn;
i dx, = 9.B2 73 --- Prds

(16) 2. + 7384 ... Pn And proceeding in the same manner we shall ultimately have, v being the letter preceding p, dxn_1 = 3. Fun-1 Pn d£n;

(17) - Pn dx, = Pmdfni

(18) so that, multiplying together the left-hand and the right-hand members of (14), (16), ..., we have

de, dx, dxg ... dxn = .. +aB273 ... Pn.d& df, d&z ... dśn; (19) and in (x1, x2, ... Xn), replacing X1, 82, ... X, by their values in terms of Eu, &, ...n, and denoting the function thereby obtained by +($1, 62, ... €»), we have

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