Having thus defined very precisely the characters and the mode of action of the two fluids, we must now explain the mathematical consequences of this definition, in order to compare them with the pheno mena, and to see if they are exactly conformable to them. We must endeavour, above all, to find those which, being susceptible of a precise and numerical value, admit of greater rigour in their verification. But these deductions cannot be obtained but by very profound calculations, which require all the re sources of analysis • and, even with the aid of these, it is only of late that they have been established in a general and exact manner. It is to M. Poisson that this fine discovery is due. We shall take from his treatise, published in the Memoirs of the Institute of France for 1811, the precise results which calculation has made known to him; we shall borrow them as the rigorous deductions of our first definitions; and it will then only remain for us to ascertain if they agree with the facts.
We shall begin with considering a single conduct ing and insulated body charged with an excess of vitreous or of resinous electricity, and exempt from all external influence. Setting out from the consti tution assigned to the two fluids, calculation informs 'us, that the fluid introduced into this body will dif fuse itself entirely on its surface, and will there form a stratum extremely thin. This is confirmed by ob servations the most minutely exact. Calculation de termines also the interior surface of this stratum and its thickness. The exterior surface, bounded by the air, is the same with that of the body. The air is in this case to electricity, as an impenetrable vase of * given form, which contains it in its interior capacity, and resists by its pressure the tendency which it has to escape.
The interior surface is in every case but very lit tle different from the•other, the electrical stratum be ing very thin. But in order'that the electrical state of the body may remain permanent, the form of this surface must be such, that the entire stratum exert nei ther attraction nor repulsion on the.points comprised within its cavity. For, if these actions were not re duced to nothing, they would operate upon the com bined electricities of the body, would decompose part of them, and the electrical state of the body would therefore change, contrary to the state of permanency which we have supposed. The analytical condition which establishes this property, determines the form and the thickness of the stratum, which may, and, even in general must, be unequal upon the different parts of the surface of the electrified body. (See the Memoirs of the Institute of France for 1811.) If the body, for example, has the form of a sphere, the two surfaces of the electrical stratum will be spherical, and will have their centre in the centre of the sphere. The thickness of the stratum then will be everywhere constant, and equal to the difference of their radii.
Newton, indeed, has long since demonstrated, that, in the law of the square of the distance, such a stra tum exerts no action on the points which are within. (Princip. Math. Lib. 1. Prop. LXX.) If the proposed spheroid is an elipsoid, the inte rior surface of the electrical stratum will be also an , ellipsoid, concentric and similar; for it can be de monstrated, that an elliptical stratum, of which the surfaces are also concentric and similar, exerts no action on a point situated in its interior. The thick ness of the stratum in every point is determined ge nerally by this construction. It hence follows, that this thickness is greatest at the extremity of the greater axis, and least at the extremity of the small er; and the thicknesses corresponding to the two ex tremities of the different axes, are to each other as the lengths of these axes, which, as we have seen, is conform to the experiments. In general, the exterior surface of the fluid stratum is given by the surface of the body itself; and the whole problem is redu ced to find for the interior surface a form very little different from this, which shall bring to nothing the total action of the stratum on all the points compri sed within its cavity.
The electrical stratum thus disposed, acts by at traction and repulsion on the other electrical parti cles situated beyond its exterior surface, or at this surface itself. It attracts them if they are of a dif ferent nature from its own, and if they are of the same nature it repels them. This last case is that of the electrical particles which form the exterior surface of the stratum, each of these being repelled from within outwards, with a force proportional to the thickness of the stratum at that point. The par ticles situated under the surface, in the thickness of the stratum itself, suffer a similar repulsion, but weak er, as it is only proportional to the thickness which separates them from the interior surface of the stra tum, for the particles with which they are enve loped on the sale of the exterior surface, according to the form of the two strata, exert on them no action at all. All these repulsive forces gradually decreas ing, and being resisted in their effects by the exter nal air, which opposes the escape of the electrical particles, it is easy to conceive, that there must re sult a total pressure exerted against this air, and tending to drive it off'. This pressure is in a ratio compounded of the repulsive force exerted at the surface, and of the thickness of the stratum ; or, as the one of these elements is always proportional to the other, we may say that, in every point, the pres sure is proportional to the square of the thickness. It may therefore in general be variable on the sur face of electrified bodies.