The forces, then, which may produce the motion of B, are, 1. The mutual attraction or repulsion of the fluid of A upon the fluid of B. 2. The repulsion of the fluid of B on itself. But it is demonstrated in mechanics, that the mutual attractions and repul sions exerted by the particles of a system of bodies on each other, cannot impress any motion on its centre of gravity ; the,effects of this internal action then de stroy themselves upon each of the spheres - there can not result from it any motion of the 'one towards the other ; and the first kind of force, therefore, is the only one to which we need pay any attention. If the electricity is distributed uniformly over every sphere, each of them attracts or repels the other as if its whole electric mass were collected in its centre. Thus, if we call a the distance of their centres, their radii, e,e' the thicknesses of the electric strata formed upon their surfaces by the quantities of elec tricity introduced into them, the electric mass of each of them will be w being the semi. circumference of which the radius is equal to unity, and the attractive or repulsive force will be express ed by K being a coefficient which ex.
K$ presses the intensity of the force when the quantities a, e, e', are each equal to the unity of their species. This force transmits itself directly to the two spheres, in consequence of the adhesion by which they retain the electric particles. We see, from this expression, that the force must become nothing, if e or e' be no thing, that is, if the one of the two spheres be not primitively charged with electricity. During the motion it suffers no alteration but what arises from the distance, because the two spheres being supposed of a perfectly non-conducting substance, their reci procal action produces upon them no new develope ment of electricity.
In the second case, where A is a non-conductor, and B a conductor, the sphere B, suffers a decompo• sition of its natural electricities by the influence of A. The opposite electricities which result from this decomposition, unite with the new quantity which has been introduced, and dispose themselves together according to the laws of the electric equilibrium. Here the motion of B towards A may be regarded under two points of view.
Suppose, first, that without disturbing the electric equilibrium of 13, we extend over its surface an in• sulating stratum, solid, without weight, and which may remain invariably attached to it. The electri eity of B, unable to escape, will press as it were against this stratum, and, by this means, transmit to the particles of the body the firms by which it is urged. The forces which then act upon the system will be, 1. The mutual attraction or repulsion of the fluid of A on the fluid of B. 2. The repulsion of the fluid of B upon itself, a repulsion, however, which cannot produce any motion upon the centre of gravity of' B. S. The pressure of the fluid of B upon the insulating envelope, a pressure, again, which being exactly counterbalanced by the reaction of this coat ing, produces still no motion whatever. The first force, then, is still the only one to which we need pay any attention.
When the distance a, of the two spheres is very great relatively to the radii of their surfaces, the de composed electricities of B, as we have seen at page 85, are distributed almost equally over the two he mispheres situated on the side of A, and on the op. poste. In that case the actions which they suffer on the part of A are nearly equal, and destroy each other; all the force then is produced by the quan tities of external electricity, 41rele introduced into the two spheres, which, acting as if they were wholly collected in their centres, the force becomes 'When the two spheres are very far from each other, the coefficient K may be considered as con stant, and the attractive or repulsive force varies not but in consequence of a change in the distance a. But
this is only an approximation; for, to consider the matter rigorously, the electrical state of the conduct ing sphere B, varies in proportion as it approaches A, on account of the separation which this produces in its natural electricities. Hence also the recipro cal action of the two spheres ought to vary a very complicated manner, and it is probably to this that we must ascribe the error which appears in the ex periments of Coulomb, at very small distances, when calculated by the simple law of the square of the dis tance.
The supposition of an insulating envelope without weight, serves here merely to connect the electric fluid with the material particles of the body B, and we may always r • as such the little stratum of air with which • • are ordinarily enveloped, and which adheres to their surfaces. Yet the same re sult may be obtained without the aid of this inter mediary; but, in that case, we must consider the prune-tee produced upon the air by the electricities which exist at liberty in B. These electricities, in effect, as well those that have been introduced, as those that are decomposed on it, move towards the surface of B, where the air stops them by its pres sure, and prevents their escape; they dispose them selves then under this surface, as their mutual ac tion and the influence of the body A require, rest ing, for this purpose, against the air, which pre.. vents them from expanding. But, reciprocally, they press this air from within outwards,'ancl tend to fly off with a force proportional to the square of the thickness of the electric stratum in every point. De compoae these pressures in the direction of three rectangular axes of the co-ordinates .r y z, the one x being in the direction of the straight line Cc (fig. 6), which joins the centres of the two spheres, and add together all the partial sums; it will then be found, as we shall show presently, that, in the direction of the co-ordinates y and z, they amount to nothing, and there only remains, therefore, a single resulting force, directed in the straight line Cc, that is, towards the centre of the sphere A. When the spheres are very distant from each other, compared with the radii of their surfaces, the decomposed electricities of B press the external air, in opposite directions, with a ffirce nearly equal, and their effects destroy each other almost exactly. There only remains, then,. the effect of the quantities e, e' introduced into the two spheres, and from this there results an excess of pressure in the direction of the lines of the centres, e' and expressed by e9 K being a constant quanti• ty for the two spheres, that is, exactly the same as was obtained by the other method. It is evident, besides, that this expression is subject to the same limitation, since the pressures produced by the elec tric stratum against the external air, ought to vary with the quantity of natural electricity decomposed on B by the influence of A, in proportion as the two spheres approach each other.