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into a function where @ is the independent variable, having given

x = r cos 0, y = r sin 0.

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Let u be a function of two variables, x and y, so that

u = f (x, y);

du then to express and in terms of two new variables

dy

dir

qs and e, of which x and y are functions given by the equations

X = 0 (r, o), y = {(1, 0), we proceed as follows. We have

du du du du dy
dr = dxdot dydis
du du du du dy
de = dado + dy.de

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du dx du do du dr.de - dodo dy dx dy dy dx

dr.do .dr.do If 7 and 0 be given explicitly in ternis of x and y, we have at once

du du du du do
- dr.da* de da'
du du dr du do

dy - dr.dy * do dy For the successive differentials we proceed in the same manner; and if there be more than two independent variables, the only difference is that the expressions become more complicated. Such cases however seldom occur.

If the independent variables enter into multiple integrals, we cannot substitute directly the values of the original differentials in terms of the new variables, because one is supposed to vary while the others are constant. To introduce this condition we proceed as follows. Let for example there be a double integral sj v dx dy, and let

Q(r, e), y = y (r, o),

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Since x is to vary when y is constant and vice versâ, we must make dy = 0) when we wish to find dx, and dx = 0) when we wish to find dy. Taking the latter condition, we have the two simultaneous equations

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From this it follows that when dy = 0, dr = 0. Hence we have

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Substituting these values in the double integral it becomes

v (dx dy dx dy dr do.

(do dr dr do)

If we had three variables x, y, z to be transformed into three others p, q, r, we should have three equations of the form

dx = Pdp + Qdq + Rdr,
dy = Pidp + Qidq + R,dr,

dx = Pdp + Qodq + R,dr; and we should determine dw by supposing dy=0, and dx = 0, and then eliminating two of the three quantities dp, dq, dr. Supposing we eliminate the last two we have dw = Mdp, M being a function of p, q, r. From this it follows that when dx = 0, dp=0. Hence supposing y to vary while w and x are constant we have

dy = Qidq + Rdr,

0 = Q2dq + Rodr; and eliminating dr between these we have dy=Ndq, N being a function of p, q, r. It follows that when dy = 0, dq = 0, and therefore if we suppose x to vary while w and y are constant, we find dx = Rodr, so that finally

dy dx = MNR,dp dq dr. The general expression for M is complicated, and it is of little use to give it here, as the consideration of the particular conditions of any given transformation will usually give us its value more readily than a substitution in the general formula.*

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• Lagrange, Mémoires de Berlin, 1773, p. 121.

Legendre, Mémoires de l'Académie des Sciences, 1788, p. 454.

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The

the variables being the samc as in the last example.
result is

dR
dr

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This equation occurs in researches on the motion of fluids.

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