7 being the distance between p and the element do, we hence deduce

0 = &T;

this substituted in the general value of SU, before given, there arises SU, 0, and consequently 0=8U.. The result just obtained being general, and applicable to any point p" within the surface, gives immediately

and we have by substituting in the equation determining &V,

0 = SV.

In a preceding part of this article, we have obtained the equation

and therefore the density (p) induced on any element do, which is evidently a function of the co-ordinates x, y, z, of p, is also

such a function as will satisfy the equation 08(p): it is moreover evident, that (p) can never become infinite when p is within the surface.

It now remains to prove that the formula

shall always give V = V, for any point within the surface and infinitely near it, whatever may be the assumed value of V.

For this, suppose the point p to approach infinitely near the surface; then it is clear that the value of (p), the density of the electricity induced by p, will be insensible, except for those parts infinitely near to p, and in these parts it is easy to see, that the value of (p) will be independent of the form of the surface, and depend only on the distance p, do. But, we shall afterwards show (art. 10), that when this surface is a sphere of any radius whatever, the value of (p) is

a being the shortest distance between p and the surface, and ƒ representing the distance p, do. This expression will give an idea of the rapidity with which (p) decreases, in passing from the infinitely small portion of the surface in the immediate vicinity of p, to any other part situate at a finite distance from it, and when substituted in the above written value of V, gives, by supposing a to vanish,

V=V.

It is also evident, that the function V, determined by the above written formula, will have no singular values within the surface under consideration.

What was before proved, for the space within any closed surface, may likewise be shown to hold good, for that exterior to a number of closed surfaces, of any forms whatever, provided we introduce the condition, that V' shall be equal to zero at an infinite distance from these surfaces. For, conceive a surface at

an infinite distance from those under consideration; then, what we have before said, may be applied to the whole space within the infinite surface and exterior to the others; consequently

4 π V ' = [do V' (dT)

(5'),

where the sign of integration must extend over all the surfaces, (seeing that the part due to the infinite surface is destroyed by the condition, that V' is there equal to zero), and dw must evidently be measured from the surfaces, into the exterior space to which ' now belongs.

The form of the equation (6) remains also unaltered, and

the sign of integration extending over all the surfaces, and (p) being the density of the electricity which would be induced on each of the bodies, in presence of each other, supposing they all communicated with the earth by means of infinitely thin conducting wires.*

(6.) Let now A be any closed surface, conducting electricity perfectly, and p a point within it, in which a given quantity of electricity is concentrated, and suppose this to induce an electrical state in A; then will V, the value of the potential function arising from the surface only, at any other point p', also within it, be such a function of the co-ordinates p and p', that we may change the co-ordinates of p, into those of p', and reciprocally, without altering its value. Or, in other words, the value of the potential function at p', due to the surface alone, when the inducing electricity is concentrated in p, is equal to that which would have place at p, if the same electricity Q were concentrated in p'.

For, in consequence of the equilibrium at the surface, we have evidently, in the first case, when the inducing electricity is concentrated in p,

? + V=B;

• In connexion with the subject of this article, see a paper by Professor Thomson, Cambridge and Dublin Mathematical Journal, Vol. VI. p. 109.

r being the distance between p and do an element of the surface A, and B a constant quantity dependent upon the quantity of electricity originally placed on 4. Now the value of V at p' is

V = − f(p') do' V,

by what has been shown (art. 5); (p) being, as in that article, the density of the electricity which would be induced on the element do by a unit of electricity in p', if the surface A were put in communication with the earth. This equation gives

since ♪V=-♪2=0; the symbol 8 referring to the co-ordinates x, y, z, of p. But we know that 0=&V; where & refers in a similar way to the co-ordinates x', y', z', of p' only. Hence we have simultaneously

0=8V and 0=&V;

where it must be remarked, that the function V has no singular values, provided the points p and p' are both situate within the surface A. This being the case the first equation evidently

gives (art. 5)

- f(p) do V;

being what would become, if the inducing point p were carried to do, p' remaining fixed. Where is a function of x', y', z', and §, n,, the co-ordinates of do, whereas (p) is a function of x, y, z, §, n, 5, independent of x, y, z'; hence by the second equation

which could not hold generally whatever might be the situation P, unless we had

of

0=8V;

where we must be cautious, not to confound the present value of

7, with that employed at the beginning of this article in proving the equation 081, which last, having performed its office, will be no longer employed.

The equation 0=8'V' gives in the same way

V = - [(p') do' V';

being what becomes by bringing the point p' to any other element do' of the surface A. This substituted for V, in the expression before given, there arises

in which double integral, the signs of integration, relative to each of the independent elements do and do', must extend over the whole surface.

If now, we represent by V', the value of the potential function at p arising from the surface A, when the electricity Q is concentrated in p', we shall evidently have

where the order of integrations alone is changed, the limits re

maining unaltered: V, being what V, would become, by first bringing the electrical point p' to the surface, and afterward the point p to which V, belongs. This being done, it is clear that V

,

and V, represent but one and the same quantity, seeing that each of them serves to express the value of the potential function, at any point of the surface A, arising from the surface itself, when the electricity is induced upon it by the action of an electrified point, situate in any other point of the same surface, and hence we have evidently

V = V1,

as was asserted at the commencement of this article.