Making ra in this, and in the value of B' before given, in order to obtain those which belong to the surface, there results dB' B - 2 + = aft = afs This substituted in the general equation written above, there 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 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 a2 - b3 = Απαξ 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, = P -a 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 f being the distance P, do. This substituted in the general equation (6), art. 5, gives = (10). In the same way we shall have, when the point P is exterior to the sphere, V = b3- a3 [do (11). 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. 11.), 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 8, 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, 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, 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 be the value of the potential function without the surface, corresponding to the value VB-A within it; then, by what precedes B V=-A'; 4' being determined from A by the following equations: A = 4, (r), _¥. (r) = = $. (—), 4'=v,(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 evidently V'. Therefore by equating 'being made infinite. Having thus the value of B, 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 P will be, and hence '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'b-2br' . cos 0+r'2. equation we deduce successively, From which 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 being the part electricity on S'. arising from S, and the total quantity of 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= 2 and F=2. 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 |