The third case, in which A and B are both con ductors, is resolved exactly upon the same princi ples, either by imagining the two electrified surfaces covered with an insulating envelope, and calculating the reciprocal actions of the two fluids which are transmitted by means of this cover to the material particles; or in considering the pressures produced on the external air by the two electric strata, and cal culating the excess of these pressures in the direc tion of the line which joins the two centres; only, in this case, the attractive or repulsive force of these two spheres will vary in proportion as they approach each other, not only by the difference which thence arises in the intensity of the electric action, but still farther by the decomposition of the natural electrici ties which will be going on in the two conducting bo dies A and B.
To render the mathematical exactness of these con siderations evident, we shall, for the preceding case, go through the calculation of the pressures exerted against the air by the quantities of electricity intro.. duced or developed on the two spheres. This will have, besides, the advantage of giving a new applica tion to the formula stated at p. 84. For this pur pose take, first, on the sphere A, fig. 6, any point whatever which we may denote by M. The pres sure exerted at this point against the air depends on the thickness of the electric stratum there. To know this, we must put the particular value of the angle u, which corresponds to the point M, in the general expression of E given at page 84, and multiply the square of this thickness by a constant coefficient k, which will disappear of itself, when we take the relations of the pressures among each other at different points. In this manner, the pres sure for any point of either sphere calculated for the unity of the surface, will be represented by upon the first, and upon the second, E, E' being taken from the formula' stated in p. 84. We shall now de-. velope, successively, these two expressions. In the first place, as the pressure varies from one point to another with the thickness of the electric stratum, we cannot suppose it the same, but in a very small space all round the point M, a space which must be considered as a superficial element of the sphere, and which we shall call 0; thus the expression being calculated for the unity of surface, the pressure upon the small superficial element a will be This pressure acts perpendicularly to the spherical surface A, in the direction of the radius CM • de compose it then into three others, parallel to three axes of the rectangular co-ordinates x, y, z, which have their origin at the centre C; the first, x, being in the direction of the straight line, Cc, which joins the centres of the spheres, and the two others perpendicular to this line. To effect this decompo
sition, we must multiply the normal pressure by the cosines of the angles which the radius CM, forms with the co-ordinates x, y, z; that is, by since, in the formula of page 84, we have r ' r ' r' represented by r the value of the radius CM of the sphere A. We will thus have the three following component parts For this second point, the element w, and the prat-. sure will be also absolutely the same; a on ac count of the symmetry of the surface of the sphere A ; on account of the symmetrical disposition of the electricity round the axis of the co-ordinates .r, which joins the centres of the two spheres A and B; but the component force which proceeds parallel to the co-ordinates z, will be — K s V, on account of the negative sign of z; this force, and its analogous one, K being equal, and in opposite directions, will mutually destroy each other, and a similar equi librium will be equally obtained, in this kind of press sure, for all the other couples of points M, M', which correspond on the two sides of the plane of x y.
A similar process of reasoning will prove that the forces w will destroy each other two and two, upon corresponding points, taken on the two sides of the plane of x a, and of which the co-ordinates will be .r, y, z for the one, and + x, — y, ? a for the other.
It remains, then, to consider the components of the pressures, taken parallel to the co-ordinates x ; that is, parallel to the straight line which joins, the centres Cc, of the two spheres; and, indeed, from the symmetrical disposition of electricity round this straight line, it is evident that it-cannot have any mo tion but in ;his single direction; and, consequently, these components alone must produce the tendency of the two spheres towards each other.
To obtain, in the simplest manner, the sum of all these components, it must be remarked, that their general expression K contains no variable but x; for cos. u, which ehters into the value of is But we must observe, first, that it is absolutely of no use paying any attention to the two last, because the efforts which each of them makes, on the whole extent of the surface, mutually destroy each other on account of the symmetrical disposition of the electricity rela tively to the axis of the co-ordinates x, whichjoins the two centres. If we consider, in effect, the force, for example, for the point M, situated in the figure under the plane of the co-ordinates x y; we shall find, above this plane, another point M' situated quite similarly, and of which the co-ordinates x, y, z, will consequently be the same, with this only difference, that z will there be negative, on account of its oppo site situation relative to the origin of the co-ordinates.