potential function at A's centre, arising from the whole exterior space, will be

pdxdydz

and the value of the same function at B's centre will be

pdxdydz

the integrals extending over all the space exterior to the conducting system under consideration.

If now, Q be the total quantity of electricity on A's surface, and that on B's, their radii being a and a'; it is clear, the value of the potential function at A's centre, arising from the system itself, will be

seeing that, we may neglect the part due to the wire, on account of its fineness, and that due to the other sphere, on account of its distance. In a similar way, the value of the same function at B's centre will be found to be

옴.

But (art. 1) the value of the total potential function must be constant throughout the whole interior of the conducting system, and therefore its value at the two centres must be equal; hence

Although p, in the present case, is exceedingly small, the integrals contained in this equation may not only be considerable, but very great, since they are of the second dimension relative to space. The spheres, when at a great distance from each other, may therefore become highly electrical, according to the observations of experimental philosophers, and the charge they will receive in any proposed case may readily be calculated; the value of p being supposed given. When one of the spheres,

B for instance, is comected with the ground, Q' will be equal to zero, and consequently Q immediately given. If, on the contrary, the whole system were insulated and retained its natural quantity of electricity, we should have, neglecting that on the wire,

0 = Q + Q',

and hence Q and Q' would be known.

If it were required to determine the electrical state of the sphere 4, when in communication with a wire, of which one extremity is elevated into the atmosphere, and terminates in a fine point p, we should only have to make the radius of B, and consequently, Q', vanish in the expression before given. Hence in this case

r' being the distance between p and the element dxdydz. Since the object of the present article is merely to indicate the cause of some phenomena of atmospherical electricity, it is useless to extend it to a greater length, more particularly as the extreme difficulty of determining correctly the electrical state of the atmosphere at any given time, precludes the possibility of putting this part of the theory to the test of accurate experiment.

(12.) Supposing the form of a conducting body to be given, it is in general impossible to assign, rigorously, the law of the density of the clectric fluid on its surface in a state of equilibrium, when not acted upon by any exterior bodies, and, at present, there has not even been found any convenient mode of approximation applicable to this problem. It is, however, extremely easy to give such forms to conducting bodies, that this law shall be rigorously assignable by the most simple means. The following method, depending upon art. 4 and 5, seems to give to these forms the greatest degree of generality of which they are susceptible, as, by a tentative process, any form whatever might be approximated indefinitely.

Take any continuous function V', of the rectangular coordinates x', y', z', of a point p', which satisfies the partial differential equation 08V', and vanishes when p' is removed to an infinite distance from the origin of the co-ordinates.

Choose a constant quantity b, such that V'=b may be the equation of a closed surface A, and that V' may have no singular values, so long as p' is exterior to this surface: then if we form a conducting body, whose outer surface is A, the density of the electric fluid in equilibrium upon it, will be represented by

and the potential function due to this fluid, for any point p', exterior to the body, will be

hV';

h being a constant quantity dependent upon the total quantity of electricity Q, communicated to the body. This is evident from what has been proved in the articles cited.

Let R represent the distance between p', and any point within A; then the potential function arising from the elecQ tricity upon it will be expressed by, when R is infinite. Hence the condition

which will serve to determine h, when Q is given.

In the application of this general method, we may assume for V', either some analytical expression containing the coordinates of p', which is known to satisfy the equation 0 = 8V', and to vanish when p' is removed to an infinite distance from the origin of the co-ordinates; as, for instance, some of those given by LAPLACE (Méc. Céleste, Liv. 3, Ch. 2), or, the value a potential function, which would arise from a quantity of electricity anyhow distributed within a finite space, at a point p' without that space; since this last will always satisfy the conditions to which V' is subject.

It may be proper to give an example of each of these cases. In the first place, let us take the general expression given by LAPLACE,

U (0)

U (1)

U

=- + + + &c.,

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then, by confining ourselves to the two first terms, the assumed value of V' will be

r being the distance of p' from the origin of the co-ordinates, and U", U", &c. functions of the two other polar co-ordinates and . This expression by changing the direction of the axes, may always be reduced to the form

2a k2 cos 0 =- +

a and k being two constant quantities, which we will suppose positive. Then if b be a very small positive quantity, the form of the surface given by the equation V'=b, will differ but little from a sphere, whose radius is :

2a

by gradually increasing 6, the difference becomes greater, until 6; and afterwards, the form assigned by V-b, becomes improper for our purpose.

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Making therefore b= in order to have a surface differing as

much from a sphere, as the assumed value of V' admits, the equation of the surface A becomes

If now represents the angle formed by dr and do', we have

and as the electricity is in equilibrium upon A, the force with which a particle p, infinitely near to it, would be repelled, must

be directed along du': but the value of this force is

dv

dro

and

consequently its effect in the direction of the radius r, and tend

the horizontal lines over quantities, indicating, as before, that

they belong to the surface itself. The value of

duced from this equation, is

de

dw'

this substituted in the general value of p, before given, there

Supposing Q is the quantity of electricity communicated to the surface, the condition

before given, becomes, sincer may here be substituted for R, seeing that it is measured from a point within the surface,

We have thus the rigorous value of p for the surface A whose