Take any continuous function V', of the rectangular coordinates a', y', z', of a point p', which satisfies the partial differential equation ( = 8V', and vanishes when p' is removed to an infinite distance from the origin of the co-ordinates. Choose a constant quantity b, such that V'=b may be the equation of a closed surface A, and that V' may have no singular values, so long as p' is exterior to this surface: then if we form à conducting body, whose outer surface is A, the density of the electric fluid in equilibrium upon it, will be represented by -h dv' p=4t dro' and the potential function due to this fluid, for any point p', exterior to the body, will be hV'; h being a constant quantity dependent upon the total quantity of electricity Q, communicated to the body. This is evident from what has been proved in the articles cited. Let R represent the distance between p', and any point within A; then the potential function arising from the electricity upon it will be expressed by , when R is infinite. Hence the condition = RV' (R being infinite), which will serve to determine h, whenQ is given. In the application of this general method, we may assume for V', either some analytical expression containing the coordinates of p', which is known to satisfy the equation () = 8V', and to vanish when p is renoved to an infinite distance from the origin of the co-ordinates ; as, for instance, some of those given by LAPLACE (Méc. Céleste, Liv. 3, Ch. 2), or, the value a potential function, which would arise from a quantity of electricity anyhow distributed within a finite space, at a point p' without that space; since this last will always satisfy the conditions to which V' is subject. It may be proper to give an example of each of these cases. In the first place, let us take the general expression given by LAPLACE, then, by confining ourselves to the two first terms, the assumed value of V' will be U19 (1) V's qr being the distance of p' from the origin of the co-ordinates, and U", U", &c. functions of the two other polar co-ordinates O and w. This expression by changing the direction of the axes, may always be reduced to the form V-2a, kcos * goli a and ke being two constant quantities, which we will suppose positive. Then if 6 be a very small positive quantity, the form of the surface given by the equation V'=b, will differ but little from a sphere, whose radius is 24: by gradually increasing b, the difference becomes greater, antil b = ; and afterwards, the form assigned by V=b, becomes improper for our purpose. Making therefore b= , in order to have a surface differing as much from a sphere, as the assumed value of V' admits, the equation of the surface A becomes , 2a Rene cos 0 0 y' = If now $ represents the angle formed by dr and dw', we have this force is di and as the electricity is in equilibrium upon. A, the force with which a particle p, infinitely near to it, would be repelled, must be directed along dw': but the value of this force is consequently its effect in the direction of the radius r, and tending to increase it, will be a cos 6. This last quantity is equally represented by a man, and therefore det -- con i the horizontal lines over quantities, indicating, as before, that they belong to the surface itself. The value of Gover), deduced from this equation, is this substituted in the general value of p, before given, there arises - đỡ ba 2 cos P = 42 dw 27y.cos . Supposing Q is the quantity of electricity communicated to the surface, the condition (where R is infinite), before given, becomes, since gu may here be substituted for R, seeing that it is measured from a point within the surface, We have thus the rigorous value of p. for the surface A whose equation is r= (1 + wa cosy when the quantity Q of elec tricity upon it is known, and by substituting for q and h their values just given, there results. Moreover the value of the potential function for the point o' whose polar co-ordinates are r, 0, and a, is 2. QK cos 6 From which we may immodiately deduce the forces acting on any point p exterior to A. In tracing the surface A, 8 is supposed to extend from 0 = 0 to A=, and w, from w=0 to w=27: it is therefore evident, by constructing the curve whose equation is that the parts about P, where 0 =T, approximate continually in form towards a cone whose apex is. P, and as the density of the electricity at P is null, in the example before us, we may make this general inference: when any body whatever has a part of its surface in the form of a cone, directed inwards; the density of the electricity in equilibrium upon it, will be null at its apex, precisely the reverse of what would take place, if it were directed outwards, for then, the density at the apex would become infinite *. * Since this was written, I have obtained formulæ serving to express, generally, the law of the distribution of the electric fluid near the apex 0 of a cono, which forms part of a conducting surface of revolution having the same axis. From these formulæ it results that, when the apex of the cone is directed inwards, the density of the electric fluid at any point p, near to it, is proportional to guid-1; je being the distance Op, and the exponent n very nearly such as would satisfy the simple equation (4n+2) B=37 : where 2ß is the angle at the summit of the cone. If 28 exceeds t, this summit is directed outwarda, and when the excess is not very considerable, n will be given as above : but 28 still increasing, until it becomes 27 - 27; the angle zy at the summit of the cone, which is now directed outwards, being very small, n will be given by an log =1, and in case the cenducting body is a sphere whose radius is b, on which Prepreseuts the mean density As a second example, we will assume for V', the value of the potential function arising from the action of a line uniformly covered with electricity. Let 2a.be the length of the line, y the perpendicular falling from any point p upon it, ac the distance of the foot of this perpendicular from the middle of the line, and ac' that of the element dac' from the same point: then taking the element da', as the measure of the quantity of electricity it contains, the assumed value of V' will be Q-W+W{Y* + (a - x)} Wy+ (x -- x')"} -a-+ {y* + (a + 2)"} the integral being taken from a = -a to a' = +a. Making this equal to a constant quantity log b, we shall have, for the equation of the surface A, a-*+ {y* + (a - x)"} , -a-x+v{y* + (a + x)"} which by reduction becomes 0 = y* (1 – 6)* + xo. 46 (1 – 6)* -- a*. 4b (1+b). We thus see that this surface is a spheroid produced by the revolution of an ellipsis about its greatest diameter; the semitransverse axis being and semi-conjugate an-bay. By differentiating the general value of V', just given, and substituting for y its value at the surface A, we obtain of the electric fluid, p, the value of the density near the apex 0, will be determined by the formula u being the length of the cone. |