x', y', z', and that whose co-ordinates are x, y, z, to which p' belongs, also r the distance between p' and do, 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 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 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 8V,=0 and 8 1 where dw' is measured from the surface into the exterior space to which V, belongs, as dw is, into the interior space. Consequently dwdw', and therefore do V == do V = do (dV Hence the equation determining p becomes, by substituting for Thus the whole difficulty is reduced to finding the value V, 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 formulæ, however great that number may be: V, 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 RESULTS 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 the thickness of the glass measured along a normal to A; 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 C, 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 8. Moreover, the same potential function within the substance of the outer coating, and in the interior of the conductor C', will be equal to another constant quantity B'. 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; we shall have (art. 4) V=ẞ and VB'. 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 axes 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 w, w' and w", and we shall have by TAYLOR's theorem, since w' = 0 and w" = 0 at the axis of w along which is measured, V = V + ᏧᏙ Ꮎ d2 V 02 + dw 1 dr2 1.2 + &c.; where, on account of the smallness of 0, the series converges very rapidly. By writing in the above, for V and their values just given, we obtain In the same way, if w be a normal to B, directed towards A, and 0, be the thickness of the glass measured along this normal, we shall have But, if we neglect quantities of the order e, compared with those retained, the following equation will evidently hold good, n being any whole positive number, the factor (− 1)" being introduced because w and w are measured in opposite directions. = p and being the densities of the electric fluid at the surfaces A and B respectively. Permitting ourselves, in what follows, to neglect quantities of the order 2 compared with those retained, it is clear that we may write 0 for 0,, and hence by substitution where V and p are quantities of the order; ' and 3 being the order or unity. The only thing which now remains to 0° d2 V be determined, is the value of for any point on the surface A. Throughout the substance of the glass, the potential function will satisfy the equation 0= SV, and therefore at a point on the surface of A, where of necessity w, w' and w" are each equal to zero, we have 12 dw2 the horizontal mark over w, w' and w" being, for simplicity, omitted. Then since w' = 0, and as V is constant and equal to ẞ at the surface A, there hence arises R being the radius of curvature at the surface A, in the plane (w, w'). Substituting these values in the expression immediately preceding, we get |