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we find

when 2 + y2+ x2 = r2,

φ

dr2

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d2 V

= 0.

(5) Transform + = 0 into a function of r and

dx2 dy

0, having given ar cos 0, y = r sin 0.

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into a function of r, 0, and p, having given

x = r cos 0, y = r sin 0 sin 4, x = r sin 0 cos p.

A slight artifice will enable us to do this with considerable facility. Assume pr sin 0,

so that

x = p cos &,

y = p sin o,

p = r sin 0,

x = r cos 0.

Taking first the two variables y and x, we find as in the preceding example

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In exactly the same way, the equations of condition being similar, we find

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Also, as in the first part of the last example,

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By substituting for p its value, and making some obvious reductions, this becomes

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This important equation is the basis of the Mathematical

Theories of Attraction and Electricity.

The artifice here

used is given by Mr A. Smith in the Cambridge Mathematical Journal, Vol. 1. p. 122.

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into one where u and v are the independent variables, x, y, u, v being connected by the equations

x + y = u, y = UV.

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Therefore dy dx = udud v,

and ffx-1y-1 dy dx = ffum+n-1 (1 − v)m−1 y2−1 du dv.

This transformation is given by Jacobi in Crelle's Journal, Vol. XI. p. 307: it is of great use in the investigation of the values of definite integrals.

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into one where r and are the independent variables, having given

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

[Sex2+ y2 dx dy = - Sfer3r dr do.

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Using the same artifice as in Ex. 6, we find

[[[V dx dy dx = [[[Vr2 dr sin ✪ de dø.

This is a very important transformation, being that from rectangular to polar co-ordinates in space.

If we suppose V = 1, fffdx dy do is the expression for the volume of any solid referred to rectangular co-ordinates: and it becomes fffr2 dr sine de dp when referred to polar co-ordinates.

(10) Having given a function of x and y determined. by the equation

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= ffd0d & sin 0 { a2b2 (cos 0)2 + (c sin()2 (a2 sin2 + b2 cos ̊p)}1.

Ivory, Phil. Trans. 1809.

CHAPTER IV.

ELIMINATION OF CONSTANTS AND FUNCTIONS BY MEANS OF DIFFERENTIATION.

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To eliminate a, substitute its value given by (2) in (1);

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To eliminate both a and b, differentiate (2) again; then

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(4) Eliminate a and b from the equation

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