Making ra in this, and in the value of B' before given, in order to obtain those which belong to the surface, there results

This substituted in the general equation written above, there arises

If P is supposed to approach infinitely near to the surface, so that b=a-a; a being an infinitely small quantity, this would become

P

a

In the same way, by the aid of the equation between A and p, the density of the electric fluid, induced on the surface of a sphere whose radius is a, when the electrified point P is exterior to it, is found to be

supposing the sphere to communicate, by means of an infinitely fine wire, with the earth, at so great a distance, that we might neglect the influence of the electricity induced upon it by the action of P. If the distance of P from the surface be equal to an infinitely small quantity a, we shall have in this case, as in the foregoing,

From what has preceded, we may readily deduce the general value of V, belonging to any point P, within the sphere, when Vits value at the surface is known. For (p), the density induced upon an element do of the surface, by a unit of electricity concentrated in P, has just been shown to be

b2-a2
;

f being the distance P, do. This substituted in the general equation (6), art. 5, gives

In the same way we shall have, when the point P is exterior to

The use of these two equations will appear almost immediately, when we come to determine the distribution of the electric fluid, on a thin spherical shell, perforated with a small circular orifice.

The results just given may be readily obtained by means of LAPLACE'S much admired analysis (Méc. Cél. Liv. 3, Ch. II.), and indeed, our general equations (9), flow very easily from the equation (2) art. 10 of that Chapter. Want of room compels me to omit these confirmations of our analysis, and this I do the more freely, as the manner of deducing them must immediately occur to any one who has read this part of the Mécanique Céleste.

Conceive now, two spheres S and S', whose radii are a and a, to communicate with each other by means of an infinitely fine wire: it is required to determine the ratio of the quantities of electric fluid on these spheres, when in a state of equilibrium; supposing the distance of their centres to be represented by b.

The value of the potential function, arising from the electricity on the surface of S, at a point p, placed in its centre, is

do being an element of the surface of the sphere, p the density of the fluid on this element, and Q the total quantity on the sphere. If now we represent by F", the value of the potential function for the same point p, arising from S', we shall have, by adding together both parts,

F+ ;

the value of the total potential function belonging to p, the centre of S. In like manner, the value of this function at p', the centre of S', will be

F+

Q

α

F being the part arising from S, and the total quantity of electricity on S'. But in consequence of the equilibrium of the system, the total potential function throughout its whole interior is a constant quantity. Hence

Although it is difficult to assign the rigorous values of F and F'; yet when the distance between the surfaces of the two spheres is considerable, compared with the radius of one of them, it is easy to see that F and F' will be very nearly the same, as if the electricity on each of the spheres producing them was concentrated in their respective centres, and therefore we have very nearly

F

Q
= and F

Q'

b

Thus the ratio of Q to Q' is given by a very simple equation, whatever may be the form of the connecting wire, provided it be

a very

fine one.

If we wished to put this result of calculation to the test of experiment, it would be more simple to write P and P' for the mean densities of the fluid on the spheres, or those which would be observed when, after being connected as above, they were separated to such a distance, as not to influence each other sensibly. Then since

We therefore see, that when the distance b between the centres of the spheres is very great, the mean densities will be inversely as the radii; and these last remaining unchanged, the density on the smaller sphere will decrease, and that on the larger increase in a very simple way, by making them approach each other.

Lastly, let us endeavour to determine the law of the distribution of the electric fluid, when in equilibrium on a very thin spherical shell, in which there is a small circular orifice. Then, if we neglect quantities of the order of the thickness of the shell, compared with its radius, we may consider it as an infinitely thin spherical surface, of which the greater segment S is a perfect conductor, and the smaller one s constitutes the circular orifice. In virtue of the equilibrium, the value of the potential function, on the conducting segment, will be equal to a constant quantity, as F, and if there were no orifice, the corresponding value of the density would be

F
Απα

a being the radius of the spherical surface. Moreover on this supposition, the value of the potential function for any point P,

F

within the surface, would be F. Let therefore, +p re

Απα

present the general value of the density, at any point on the surface of either segment of the sphere, and F+V, that of the corresponding potential function for the point P. The value of the potential function for any point on the surface of the sphere will be F+V, which equated to F, its value on S, gives for the whole of this segment

0 = V.

Thus the equation (10) of this article becomes

the integral extending over the surface of the smaller segment s only, which, without sensible error, may be considered as a plane.

But, since it is evident that p is the density corresponding to the potential function V, we shall have for any point on the segments, treated as a plane,

as it is easy to see, from what has been before shown (art. 4); dw being perpendicular to the surface, and directed towards the centre of the sphere; the horizontal line always serving to indicate quantities belonging to the surface. When the point P is very near the plane s, and z is a perpendicular from P upon s, z will be a very small quantity, of which the square and higher powers may be neglected. Thus ba-z, and by substitution

the integral extending over the surface of the small plane s, and f being, as before, the distance P, do. Now

provided we suppose z = 0 at the end of the calculus. Now the

F

density +p, upon the surface of the orifices, is equal to

Απα

zero, and therefore we have for the whole of this surface

the integral extending over the whole of the plane s, of which do is an element, and z being supposed equal to zero, after all the operations have been effected.