For the particular case in which the two electri fied spheres are removed to a great distance from each other, in relation to the radius of any one of them, M. Poisson has discovered formula; which ex press in a very simple manner' the thickness of the electric stratum, in any point of their surfaces. We shall here state these formula:, as they enable us to explain distinctly why conducting bodies, when they are electrified, seem to attraFt or repel each other, although, from the manner in which electricity is distributed among them, and from its mobility in their interior, we cannot suppose that these pheno mena indicate any sensible affinity which it has for their substance. Let r,r' represent the radii of the two spheres; call e,e' the thicknesses of the strata which the quantities of electricity they possess would form upon their surfaces if they were left to them selves, and exempt from all external influence; call a the distance of their centres, and place them so far from each other that the radius r' of one of them be very small compared with a, and with a r. Lastly, let u,u' denote the angles formed with the distance a, by the radii drawn from the centre of each sphere to any point on their surfaces ; then. the thicknesses E,E' of the electric stratum in these points will be expressed approximately by the fol lowing fonnulte.
Here, as in the experiments of Coulomb, the angles are reckoned from the points A and a (fig. 5), in which the surfaces of the two spheres would touch each other, if we brought them to the point of contact. The difference of symmetry in these expressions is owing to this, that the approximation from which they arise supposes the radius r', of the second, very small compared with the distance a r, which separates its centre from the surface of the other.
If it is required, for example, from these formulae to determine the state of an insulated, but not elec trified sphere, which we present to the influence of another sphere charged with a certain quantity of electricity, we have only to suppose e' nothing in the equation of the second sphere, and it will then become ,, E' S cos. u' (3 u' (i a At the point a, on the line Aa, between the two centres, the angle u' is nothing. In this point then 3 2 5 r'we have cos. u' = 1: and E' er (1 3 a) The thickness E', then, has always a contrary sign to that of e, that is to say, that the electricity on this point, in the sphere of which the radius is r', is of a nature contrary to that which covers the sphere of which the radius is r.
At the point d, diametrically opposite to the pre ceding, the angle u' is equal to 180°, which gives e 5 r' cos. u' = 1; and E'= 3 ( 3 a This value of E' has always the same sign with r' that of e; for the factor - 5 a is a fraction far smaller 3 than unity, since the distance a is supposed very great, compared with the radius r'; then the elec tric stratum will be in this point of the same nature as upon the other sphere.
Thus we see arising out of' the theory the import ant result which we have until now only established by experiment, that while a sphere c, not electrified, is placed in presence of another sphere C, electrified vitreously, for example, the combined electricities of c are partly decomposed the resinous electricity that results flowing towards the part of c which is nearest to C, and the vitreous electricity towards the part which is flirthest from it.
The thicknesses of the stratum in these two points are to each other in the ratio of 1 to 1 a 3 a ' they are nearly equal, then, since a is supposed very great in relation to r'.
Hence it may be conceived that there must be upon the sphere c a series of points, in which the thickness of the electric stratum is nothing, and which form a curve of separation between the two fluids. The locus of these points will be found by putting the general expression of the thickness E' equal to zero, which gives the condition If the distance a were altogether infinite, compar ed with the radius r', the second member of this equation would be reduced to cos. u'; consequently, this cosine would be 0, which would give u' = 90°. The line of separation of the two fluids would then be the circumference of the great circle, of which the plane is perpendicular to the line of tole centres.
But if a is not infinite, it is at least very great re 5r" latively to r'. Thus, the factor will still be a 6a very small fraction, and-the true value of cos. a' will be equally so. We may, therefore, in calculating, 5r' neglect the product of 6a by compared with the product of this same quantity by unity. With this modification the equation resolves itself and gives Cos. u' 571 6a In this case the line of separation of the two fluids is still a circle whose plane is perpendicular to the line of the centres but the distance of this plane from the centre of the sphere, in place of being no 57" 2 thing, is equal to r' cos. u' or , this distance be 6a ing taken from c to a, towards the electrified sphere C.
In considering only the degree of the equation which determines generally cos. u', there would seem to be two values of this cosine which would satisfy the conditions of our problem ; but it will clearly appear, that one of those roots should necessarily be greater than unity, and consequently, will not have here any real application, as it would correspond to an arc u', which is imaginary.