It 18 evident from art. 5, that our preceding arguments will be equally applicable to the space exterior to the surfaces of any number of conducting bodies, provided we introduce the condition, that the potential function V, belonging to this space, shall be equal to zero, when either p or p' shall remove to an infinite distance from these bodies, which condition will evidently be satisfied, provided all the bodies are originally in a natural state. Supposing this therefore to be the case, we see that the potential function belonging to any point p' of the exterior space, arising from the electricity induced on the surfaces of any number of conducting bodies, by an electrified point in p, is equal to that which would have place at p, if the electrified point were removed to p'. What has been just advanced, being perfectly independent of the number and magnitude of the conducting bodies, may be applied to the case of an infinite number of particles, in each of which the fluid may move freely, but which are so constituted that it cannot pass from one to another. This is what is always supposed to take place in the theory of magnetism, and the present article will be found of great use to us when in the sequel we come to treat of that theory. (7.) These things being established with respect to electrified surfaces; the general theory of the relations between the density of the electric fluid and the corresponding potential functions, when the electricity is disseminated through the interior of solid bodies as well as over their surfaces, will very readily flow from what has been proved (art. 1). For this let V represent the value of the potential function at a point p\ within a solid body of any form, arising from the whole of the electric fluid contained in it, and p be the density of the electricity in its interior; p' being a function of the three rectangular co-ordinates x,y,z: then if p be the density at the surface of the body, we shall have r being the distance between the pointy whose co-ordinates are x, y', z', and that whose co-ordinates are x, y, z, to which p belongs, also r the distance between p and da, an element of the surface of the body: V being evidently a function of x, y, z. If now V be what V becomes by changing x, y, z into x^ y, z, it is clear from art. 1, that p will be given by 0 = 4717/+87. Substituting for p', the value which results from this equation, in that immediately preceding we obtain which, by means of the equation (3) art. 3, becomes / r 4ir{ j^^^dto^ j r ^dw^' the horizontal lines over the quantities indicating that they belong to the surface itself. Suppose V, to be the value of the potential function in the space exterior to the body, which, by art 5, will depend on the value of V at the surface only; and the equation (2) art. 3, applied to this exterior space, will give, since 8F, = 0 and 8 - = 0, r where dw is measured from the surface into the exterior space to which V, belongs, as dw is, into the interior space. Consequently dw = — dw, and therefore its value just given, fpda-i rda ((W\/Jv\) J r 47r J r \\dw J \dw')y an equation which could not subsist generally, unless -1 [dV AdV\ ... P'J^fa+dV) (7) Thus the whole difficulty is reduced to finding the value Vt of the potential function exterior to the body. Although we have considered only one body, it is clear that the same theory is applicable to any number of bodies, and that the values of p and p will be given by precisely the same formulae, however great that number may be: Vt being the exterior potential function common to all the bodies. In case the bodies under consideration are all perfect conductors, we have seen (art. 1), that the whole of the electricity will be carried to their surfaces, and therefore there is here no place for the application of the theory contained in this article; but as there are probably no perfectly conducting bodies in nature, this theory becomes indispensably necessary, if we would investigate the electrical phenomena in all their generality. Having in this, and the preceding articles, laid down the most general principles of the electrical theory, we shall in what follows apply these principles to more special cases; and the necessity of confining this Essay within a moderate extent, will compel us to limit ourselves to a brief examination of the more interesting phenomena. APPLICATION OF THE PRECEDING EESULTS TO THE THEORY OF ELECTRICITY. (8.) The first application we shall make of the foregoing principles, will be to the theory of the Leyden phial. For this, we will call the inner surface of the phial A, and suppose it to be of any form whatever, plane or curved, then, B being its outer surface, and 6 the thickness of the glass measured along a normal to A; 6 will be a very small quantity, which, for greater generality, we will suppose to vary in any way, in passing from one point of the surface A to another. If now the inner coating of the phial be put in communication with a conductor G, charged with any quantity of electricity, and the outer one be also made to communicate with another conducting body C, containing any other quantity of electricity, it is evident, in consequence of the communications here established, that the total potential function, arising from the whole system,, will be constant throughout the interior of the inner metallic coating, and of the body C. We shall here represent this constant quantity by Moreover, the same potential function within the substance of the outer coating, and in the interior of the conductor G', will be equal to another constant quantity 0'. Then designating by V, the value of this function, for the whole of the space exterior to the conducting bodies of the system, and consequently for that within the substance of the glass itself j we shall have (art. i) F=/9 and f**0. One horizontal line over any quantity indicating that it belongs to the inner surface A, and two showing that it belongs to the outer one B. At any point of the surface A, suppose a normal to it to be drawn, and let this be the ax^s of w: then w, w", being two other rectangular axes, which are necessarily in the plane tangent to A at this point; V may be considered as a function of to, w and w", and we shall have by Taylor's theorem, since w = 0 and w" = 0 at the axis of w along which 6 is measured, == == dV 6 d%V g where, on account of the srnallness of 0, the series converges very rapidly. By writing in the above, for V and F their values just given, we obtain dV 0 d*V P In the same way, if w be a normal to B, directed towards A, and 0t be the thickness of the glass measured along this normal, we shall have But, if we neglect quantities of the order 9, compared with those retained, the following equation will evidently hold good, <LL=triy±L. dwn dwn n being any whole positive number, the factor (— 1)" being introduced because vi and v> are measured in opposite directions. Now by article 4 -a? , = df |