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2. Fa, b2 ... Prdens

And proceeding by processes similar to the above,

dx, =

2. + a, b, ... r, and therefore by substitution

Anzota, ß2 ... Pn de dx, dx , ... dx, = (-)" 2 --- ** df, df2 ... den: (28)

2. + a, b, ... rnus Equation (19) is evidently only a particular case of (28); that viz., in which a, = b2 = cg = ... = rn = 1, and all the other partial derived functions vanish.

Hence we have the following theorem ;
In transforming the multiple integral

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(29)

ana), Cenas).
Conta), )....

Cara), Cena)....
llena, )......
ceni, cene .. Como
(), Ces)... (e)

(30)

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.

Ex. 1. Take the double integral/ | F(x, y) dy dx ; and (1) let the equations of transformation be

X = 6 cos a-n sin a, y=&sin a + n cos a.
.. dx = cos a df-sin a dn,

dy = sin a df+cos a dn;
... a=1, 4, = (cos a)2 + (sin a)2 = 1.
i dy dx = dn .

(31) (2) Let the equations of transformation be

x = r cos 0, y = r sin 0.
... dx = cos 0 dr-r sin 0 do,

dy = sin 0 dr +r cos 0 do;S
.. a=1, 4, =r(cos 0)2 +r (sin 0)2 = r;
is dy dx = r dr do.

(32) 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 = fi(,n), y = fg (8,); then

(33)

de = ( de las) of+ a ) dan 1

dy = (alfa) s + cenadan ; } :: de dy = + {care o f a real )} dę dn. (34) Ex. 2. Let the integral be the triple integral

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Ex. 5. To transform into its equivalent dx dy dz, when x=lr, y = mr, 2 = nr; l, m, and n being subject to the condition

12+ m2 + m2 = 1. By a process similar to those above, we have the following results: 2 ga dr dm dn gol dr dn di pe dr dl dm

(45) m

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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

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