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THE object of this Essay is to submit to Mathematical Analysis the phenomena of the equilibrium of the Electric and Magnetic Fluids, and to lay down some general principles equally applicable to perfect and imperfect conductors; but, before entering upon the calculus, it may not be amiss to give a general idea of the method that has enabled us to arrive at results, remarkable for their simplicity and generality, which it would be very difficult if not impossible to demonstrate in the ordinary way.
It is well known, that nearly all the attractive and repul. sive forces existing in nature are such, that if we consider any material point p, the effect, in a given direction, of all the forces acting upon that point, arising from any system of bodies 8 under consideration, will be expressed by a partial differential of a certain function of the co-ordinates which serve to define the point's position in space. The consideration of this function is of great importance in many inquiries, and probably there are none in which its utility is more marked than in those about to engage our attention. In the sequel we shall often have occasion to speak of this function, and will therefore, for abridgement, call it the potential function arising from the system S. If p be a particle of positive electricity under the influence of forces arising from any electrified body, the function in question, as is well known, will be obtained by dividing the quantity of electricity in each element of the body, by its distance from the particle p, and taking the total sum of these quotients for the whole body, the quantities of electricity in those elements which are negatively electrified, being regarded as negative.
electricity under ction in question of electricity
It is by considering the relations existing between the density of the electricity in any system, and the potential functions thence arising, that we have been enabled to submit many electrical phenomena to calculation, which had hitherto resisted the attempts of analysts; and the generality of the consideration here employed, ought necessarily, and does, in fact, introduce a great generality into the results obtained from it. There is one consideration peculiar to the analysis itself, the nature and utility of which will be best illustrated by the following sketch.
Suppose it were required to determine the law of the distribution of the electricity on a closed conducting surface A without thickness, when placed under the influence of any electrical forces whatever: these forces, for greater simplicity, being reduced to three, X, Y, and 2, in the direction of the rectangular co-ordinates, and tending to increase them. Then p representing the density of the electricity on an element do of the surface, and go the distance between do and p, any other point of the surface, the equation for determining e which would be employed in the ordinary method, when the problem is reduced to its simplest form, is known to be
the first integral relative to do extending over the whole surface A, and the second representing the function whose complete differential is Xdx + Ydy + Zdz, x, y and being the co-ordinates of p.
This equation is supposed to subsist, whatever may be the position of p, provided it is situate upon A. But we have no general theory of equations of this description, and whenever we are enabled to resolve one of them, it is because some consideration peculiar to the problem renders, in that particular case, the solution comparatively simple, and must be looked upon as the effect of chance, rather than of any regular and scientific procedure.
We will now take a cursory view of the method it is proposed to substitute in the place of the one just mentioned.
Let us make B=f(xdx + Ydy + Zdz) whatever may be the position of the point p, V=foo when p is situate any where within the surface A, and V'= pdo when p is exterior to it : the two quantities V and V', although expressed by the same definite integral, are essentially distinct functions of x, y, and 2, the rectangular co-ordinates of p; these functions, as is well known, having the property of satisfying the partial differential equations
dy do V dV
If now we could obtain the values of V and V' from these equations, we should have immediately, by differentiation, the required value of p, as will be shown in the sequel.
In the first place, let us consider the function V, whose value at the surface A is given by the equation (a), since this may be written
the horizontal line over a quantity indicating that it belongs to the surface A. But, as the general integral of the partial differential equation ought to contain two arbitrary functions, some other condition is requisite for the complete determination of V. Now since V = pao, it is evident that none of its differential coefficients can become infinite when p is situate any where within the surface A, and it is worthy of remark, that this is precisely the condition reqnired: for, as will be afterwards shown, when it is satisfied we shall have generally
V=-6) do V;
the integral extending over the whole surface, and 6) being a quantity dependent upon the respective positions of p and do.
All the difficulty therefore reduces itself to finding a function V which satisfies the partial differential equation, becomes equal to the known value of V at the surface, and is moreover such that none of its differential coefficients shall be infinite when p is within A.
In like manner, in order to find V', we shall obtain V', its value at A, by means of the equation (a), since this evidently becomes
a=T' – B, i.e. T'=T. Moreover it is clear, that none of the differential coefficients of V' = pdo can be infinite when p is exterior to the surface A, and when p is at an infinite distance from A, D' is equal to zero. These two conditions combined with the partial differential equation in V', are sufficient in conjunction with its known value T at the surface A for the complete determination of V', since it will be proved hereafter, that when they are satisfied we shall have
the integral, as before, extending over the whole surface A, and 6) being a quantity dependent upon the respective position of p and do.
It only remains therefore to find a function V' which satisfies the partial differential equation, becomes equal to V' when p is upon the surface A, vanishes when p is at an infinite distance from A, and is besides such, that none of its differential coefficients shall be infinite, when the point p is exterior to A.
All those to whom the practice of analysis is familiar, will readily perceive that the problem just mentioned, is far less difficult than the direct resolution of the equation (a), and therefore the solution of the question originally proposed has been rendered much easier by what has preceded. The peculiar consideration relative to the differential coefficients of V and V', by restricting the generality of the integral of the partial differential equation, so that it can in fact contain only one arbitrary function, in the place of two which it ought otherwise to have contained, and, which has thus enabled us to effect the simplification in question, seems worthy of the attention of analysts, and may be of use in other researches where cquations of this nature are employed.
We will now give a brief account of what is contained in the following Essay. The first seven articles are employed in demonstrating some very general relations existing between the density of the electricity on surfaces and in solids, and the corresponding potential functions. These serve as a foundation to the more particular applications which follow them. As it would be difficult to give any idea of this part without employing analytical symbols, we shall content ourselves with remarking, that it contains a number of singular equations of great generality and simplicity, which seem capable of being applied to many departments of the electrical theory besides those considered in the following pages.
In the eighth article we have determined the general values of the densities of the electricity on the inner and outer surfaces of an insulated electrical jar, when, for greater generality, these surfaces are supposed to be connected with separate conductors charged in any way whatever; and have proved, that for the same jar, they depend solely on the difference existing between the two constant quantities, which express the values of the potential functions within the respective conductors. Afterwards, from these general values the following consequences have been deduced :
When in an insulated electrical jar we consider only the electricity accumulated on the two surfaces of the glass itself, the total quantity on the inner surface is precisely equal to that on the outer surface, and of a contrary sign, notwithstanding the great accumulation of electricity on each of them: 80 that if a communication were established between the two sides of the jar, the sum of the quantities of electricity which would manifest themselves on the two metallic coatings, after the discharge, is exactly equal to that which, before it had taken place, would have been observed to have existed on the surfaces of the coatings farthest from the glass, the only portions then sensible to the electrometer.