induce an electrical state in the shell; then will this induced state be such, that the total action on an electrified particle, placed any where within it, will be absolutely null.

For let V represent the value of the total potential function, at any point p within the shell, then we shall have at its inner durface, which is a closed one,

B being the constant quantity, which expresses the value of the potential function, within the substance of the shell, where the electricity is, by the supposition, in equilibrium, in virtue of the actions of the exterior bodies, combined with that arising from the electricity induced in the shell itself. Moreover, V evidently satisfies the equation 08, and has no singular value within the closed surface to which it belongs: it follows therefore, from Art. 5, that its general value is

V=B,

and as the forces acting upon p, are given by the differentials of V, these forces are evidently all equal to zero.

If, on the contrary, the electrified bodies are all within the shell, and its exterior surface is put in communication with the earth, it is equally easy to prove, that there will not be the slightest action on any electrified point exterior to it; but, the action of the electricity induced on its inner surface, by the electrified bodies within it, will exactly balance the direct action of the bodies themselves. Or more generally:

Suppose we have a hollow, and perfectly conducting shell, bounded by any two closed surfaces, and a number of electrical bodies are placed, some within and some without it, at will; then, if the inner surface and interior bodies be called the interior system; also, the outer surface and exterior bodies the exterior system; all the electrical phenomena of the interior system, relative to attractions, repulsions, and densities, will be the same as would take place if there were no exterior system, and the inner surface were a perfect conductor, put in communication with the earth; and all those of the exterior system will be the same, as if the interior one did not exist, and the outer surface were a perfect conductor, containing a quantity of electricity, equal to

the whole of that originally contained in the shell itself, and in all the interior bodies.

This is so direct a consequence of what has been shown in articles 4 and 5, that a formal demonstration would be quite superfluous, as it is easy to see, the only difference which could exist, relative to the interior system, between the case where there is an exterior system, and where there is not one, would be in the addition of a constant quantity, to the total potential function within the exterior surface, which constant quantity must necessarily disappear in the differentials of this function, and consequently, in the values of the attractions, repulsions, and densities, which all depend on these differentials alone. In the exterior system there is not even this difference, but the total potential function exterior to the inner surface is precisely the same, whether we suppose the interior system to exist or not.

(10.) The consideration of the electrical phenomena, which arise from spheres variously arranged, is rather interesting, on account of the ease with which all the results obtained from theory may be put to the test of experiment; but, the complete solution. of the simple case of two spheres only, previously electrified, and put in presence of each other, requires the aid of a profound analysis, and has been most ably treated by M. POISSON (Mém. de l'Institut. 1811). Our object, in the present article, is merely to give one or two examples of determinations, relative to the distribution of electricity on spheres, which on spheres, which may be expressed by very simple formulæ.

Suppose a spherical surface whose radius is a, to be covered with electric matter, and let its variable density be represented by pi then if, as in the Méc. Céleste, we expand the potential function V, belonging to a point p within the sphere, in the form

r being the distance between p and the centre of the sphere, and U), U, etc. functions of the two other polar co-ordinates of p, it is clear, by what has been shown in the admirable work just

mentioned, that the potential function V', arising from the same spherical surface, and belonging to a point p', exterior to this surface, at the distance r' from its centre, and on the radius r produced, will be

If, therefore, we make V = (r), and V' (r), the two funcwill satisfy the equation

tions & and

and the equation between 4 and y, in its first form, gives, by

differentiation,

' and 'being the characteristics of the differential co-efficients of and y, according to Lagrange's notation.

In the same way the equation in its second form yields

These substituted successively, in the equation by which p is determined, we have the following,

If, therefore, the value of the potential function be known, either for the space within the surface, or for that without it, the

value of the density p will be immediately given, by one or other of these equations.

From what has preceded, we may readily determine how the electric fluid will distribute itself, in a conducting sphere whose radius is a, when acted upon by any bodies situate without it; the electrical state of these bodies being given. In this case, we have immediately the value of the potential function arising from them. Let this value, for any point p within the sphere, be represented by A; A being a function of the radius r, and two other polar co-ordinates. Then the whole of the electricity will be carried to the surface (art. 1), and if V be the potential function arising from this electrified surface, for the same point p, we shall have, in virtue of the equilibrium within the sphere,

V+A=ß_or_V=ß −A;

B being a constant quantity. This value of V being substituted in the first of the equations (9), there results

the horizontal lines indicating, as before, that the quantities under them belong to the surface itself.

In case the sphere communicates with the earth, ẞ is evidently equal to zero, and p is completely determined by the above: but if the sphere is insulated, and contains any quantity Qof electricity, the value of 8 may be ascertained as follows: Let V' be the value of the potential function without the surface, corresponding to the value V8 A within it; then, by what precedes

=

A' being determined from A by the following equations:

A = 4, (r), V. (r) = ~ ~ þ, (~~),
Φι , A' = 4, (r'),

and r, being the radius corresponding to the point p', exterior

to the sphere, to which A' belongs. When r' is finite, we have 2. Therefore by equating

evidently V' =

r' being made infinite. Having thus the value of ß, the value of p becomes known.

P

To give an example of the application of the second equation in p; let us suppose a spherical conducting surface, whose radius is a, in communication with the earth, to be acted upon by any bodies situate within it, and B' to be the value of the potential function arising from them, for a point p' exterior to it. The total potential function, arising from the interior bodies and surface itself, will evidently be equal to zero at this surface, and consequently (art. 5), at any point exterior to it. Hence V'+B=0; V' being due to the surface. Thus the second of the equations (9) becomes

We are therefore able, by means of this very simple equation, to determine the density of the electricity induced on the surface in question.

Suppose now all the interior bodies to reduce themselves to a single point P, in which a unit of electricity is concentrated, and f to be the distance Pp': the potential function arising from and hence

P will be,

'being, as before, the distance between p' and the centre 0 of the shell. Let now b represent the distance OP, and the angle POp', then will f=-2br'. cos 0+r'. From which equation we deduce successively,