which shows, that V the value of V at the point p is given, when V its value at the surface is known. To convince ourselves that there does exist such a function as we have supposed Uto be; conceive the surface to be a perfect conductor put in communication with the earth, and a unit of positive electricity to be concentrated in the point p', then the total potential function arising from p and from the electricity it will induce upon the surface, will be the required value of U. For, in consequence of the communication established between the conducting surface and the earth, the total potential function at this surface must be constant, and equal to that of the earth itself, i. e. to zero (seeing that in this state they form but one conducting body). Taking, therefore, this total potential function for U, we have evidently v=L7, 0 = Si7, and U= - for T those parts infinitely near top. As moreover, this function has no other singular points within the surface, it evidently possesses all the properties assigned to U in the preceding proof. Again, since we have evidently V = 0, for all the space exterior to the surface, the equation (4) art. 4 gives where (p) is the density of the electricity induced on the surface, by the action of a unit of electricity concentrated in the point p. Thus, the equation (5) of this article becomes This equation is remarkable on account of its simplicity and singularity, seeing that it gives the value of the potential for any point p\ within the surface, when V, its value at the surface itself is known, together with (p), the density that a unit of electricity concentrated in p would induce on this surface, if it conducted electricity perfectly, and were put in communication with the earth. Having thus proved, that V the value of the potential function V, at any point p within the surface is given, provided its value V is known at this surface, we will now show, that whatever the value "of V may be, the general value of V deduced from it by the formula just given shall satisfy the equation 0 = 8F. For, the value of V at any point p whose co-ordinates are x, y, z, deduced from the assumed value of V, by the above written formula, is U being the total potential function within the surface, arising from a unit of electricity concentrated in the point p, and the electricity induced on the surface itself by its action. Then, since V is evidently independent of x, y, z, we immediately deduce Now the general value of U will depend upon the position of the point p producing it, and upon that of any other point p whose co-ordinates are x\ y', z', to which it is referred, and will consequently be a function of the six quantities, x, y, z, x, y', z\ But we may conceive U to be divided into two parts, one » (r being the distance pp) arising from the electricity in p, the other, due to the electricity induced on the surface by the action oi'p, and which we shall call Ut. Then since Ul has no singular values within the surface, we may deduce its general value from that at the surface, by a formula similar to the one just given. Thus where V is the total potential function, which would be produced by a unit of electricity inp', and therefore, \—r- ) is independent of the co-ordinates x, y, z, of p, to which 8 refers. Hence We have before supposed and as 8 - = 0, we immediately obtain Again, since we have at the surface itself r r being the distance between p and the element da, we hence deduce 0 = Sethis substituted in the general value of BUt before given, there arises 817, = 0, and consequently 0 = SL7.. The result just obtained being general, and applicable to any point p" within the surface, gives immediately 0=»o j- 1, and we have by substituting in the equation determining S V, 0 = SF. In a preceding part of this article, we have obtained the equation o.4.w+(i), which combined with 0 = S (-j- ], gives 0 = 8(p), and therefore the density (p) induced on any element d<r, which is evidently a function of the co-ordinates x, y, z, of p, is also such a function as will satisfy the equation 0 = 8 (p): it is moreover evident, that (p) can never become infinite whea-p is within the surface. It now remains to prove that the formula shall always give F= V, for any point within the surface and infinitely near it, whatever may be the assumed value of V. For this, suppose the point p to approach infinitely near the surface; then it is clear that the value of (p), the density of the electricity induced by p, will be insensible, except for those parts infinitely near to p, and in these parts it is easy to see, that the value of (p) will be independent of the form of the surface, and depend only on the distance p, da. But, we shall afterwards show (art. 10), that when this surface is a sphere of any radius whatever, the value of (p) is a being the shortest distance between p and the surface, and/ representing the distance p, da. This expression will give an idea of the rapidity with which (p) decreases, in passing from the infinitely small portion of the surface in the immediate vicinity of p, to any other part situate at a finite distance from it, and when substituted in the above written value of V, gives, by supposing a to vanish, It is also evident, that the function V, determined by the above written formula, will have no singular values within the surface under consideration. What was before proved, for the space within any closed surface, may likewise be shown to hold good, for that exterior to a number of closed surfaces, of any forms whatever, provided we introduce the condition, that V shall be equal to zero at an infinite distance from these surfaces. For, conceive a surface at an infinite distance from those under consideration; then, what we have before baid, may be applied to the whole space within the infinite surface and exterior to the others; consequently *r-/*r© «' where the sign of integration must extend over all the surfaces, (seeing that the part due to the infinite surfaee is destroyed by the condition, that V is there equal to zero), and dw must evidently be measured from the surfaces, into the exterior space to which V now belongs. The form of the equation (6) remains also unaltered, and the sign of integration extending over all the surfaces, and (/>) being the density of the electricity which would be induced on each of the bodies, in presence of each other, supposing they all communicated with the earth by means of infinitely thin conducting" wires.* (6.) Let now A be any closed surface, conducting electricity perfectly, and p a point within it, in which a given quantity of electricity Q is concentrated, and suppose this to induce an electrical state in A; then will V, the value of the potential function arising from the surface only, at any other point p', also within it, be such a function of the co-ordinates p and p', that we may change the co-ordinates of p, into those of p, and reciprocally, without altering its value. Or, in other words, the value of the potential function at p, due to the surface alone, when the inducing electricity Q is concentrated in p, is equal to that which would have place &tp, if the same electricity Q were concentrated \np\ For, in consequence of the equilibrium at the surface, we have evidently, in the first case, when the inducing electricity is concentrated in p, r * In connexion with the subject of this article, see a paper by Professor Thomson, Cambridge and Dublin Mathematical Journal, VoL VI. p. 109. |