=

(§1, §2, ... . . §n) ≥. + a1 ß2 · · · Pn • d§ 1 d§2... dɛn⋅ (20)

212.] Again, let us take the integral (10) to be the subject of transformation, and suppose the n equations connecting x1, x2, ... X, 1, 2,..., to be implicit, and to be of the forms

also let the several partial derived functions of these be denoted by letters according to the following scheme; viz.,

Now when 1 varies, dx2 = dx = ... = dx = 0; so that in calculation the quantity which is to replace da1, we have

now substituting this value for dr, in the given element of integration, and eliminating 1, 2, 3, ... En from that element by means of (21), it becomes a function of §1, x2, x3,... X1; and consequently 1, 3, ..., are all constant, when a varies; so that in calculating the quantity which is to replace dx2,

Equation (19) is evidently only a particular case of (28); that =... = r n = 1, and all the other

viz., in which a1 = b2 = C3= partial derived functions vanish.

Hence we have the following theorem ;

In transforming the multiple integral

into its equivalent in terms of 1, 2, ... En, where these variables are related by the n equations f1 = 0, ƒ1⁄2 = 0, ƒ1⁄2 = 0, . . . ƒ„ = 0 ;

...ƒn=0;

The sign of the result being ambiguous, because it depends on the order of integration in the transformed integral, and that order is obviously arbitrary*.

213.] The following are examples illustrative of the preceding theorem.

A full discussion of the theory of the transformation of multiple integrals will be found in a paper on the subject by M. Catalan, in the Mémoires couronnés par l'Académie de Bruxelles, Tome XIV, 1839, 1840.

These cases are obviously those of transformation from a rectangular system of axes, to another rectangular system and to a polar system respectively.

(3) More generally, if the equations of transformation are given in the form,

x = ƒ1 (§,n), y = f2 (§, n); then

.. dx dy = ±

df

(33)

{(d)(d) - (44) (d)} de dy. (31)

αξ dn

dn

Ex. 2. Let the integral be the triple integral

and (1) let the equations of transformation be those from a system of rectangular axes to another system of rectangular axes; that is, let

dx = sin cos & dr+r cos e cos o de ―r sin 0 sin dø,

These latter equations are those of transformation from a rectangular to a polar system in geometry of three dimensions.

which equation we shall find it convenient to use instead of the last of the group (39); differentiating these with respect to X1, X2,

...

Xn

we have

Differentiating (40) and (39) with respect to r, 0,,...On-1

whence ▲1 = (—)”—127" (sin 01)”—1 (sin 02)"-2...sin 0-1; so that

dx dx2... dxn

± 0,

= "-1(sin1)"-2 (sin 0)"-3...sin 0-2 dr do, do dom-1 (42)

If n =

3, we have

dx dx2 dx3 = r2 sin 01 dr dė1 de2 ;

1

which is the transformation given in the latter part of the preceding example.

Ex. 4. Let the equations of transformation be

n-1

.. dx dx2... dxn dx1 = ± §1”−1 §2"-2... En-1 d§1 d§2... dƐn. (44)

Hence we have the following transformation, when n = 2;

Ex. 5. To transform into its equivalent dx dy dz, when x= lr, y = mr, z = nr; l, m, and n being subject to the condition

12 + m2 + n2 = 1.

By a process similar to those above, we have the following

214.] If the original integral is definite, the transformed one will also be definite; for the latter is to be equivalent to the former in all respects, and consequently the values of the variables in both integrals must extend over the same district of