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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. V =7.
Moreover it is clear, that none of the differential coefficients of
V = j-— can be infinite when p is exterior to the surface A,
and when p is at an infinite distance from A, V is equal to zero. These two conditions combined with the partial differential equation in V, are sufficient in conjunction with its known value V7 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 (p) being a quantity dependent upon the respective position of p and da.
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 pointy 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 equations of this nature are employed.
We will now give a brief account of what is contained in the following Essay. The first seven articles arc 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: so 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.
If an electrical jar communicates by means of a long slender wire with a spherical conductor, and is charged in the ordinary way, the density of the electricity at any point of the interior surface of the jar, is to the density on the conductor itself, as the radius of the spherical conductor to the thickness of the glass in that point.
The total quantity of electricity contained in the interior of any number of equal and similar jars, when one of them communicates with the prime conductor and the others are charged by cascade, is precisely equal to that, which one only would receive, if placed in communication with the same conductor, its exterior surface being connected with the common reservoir. This method of charging batteries, therefore, must not be employed when any great accumulation of electricity is required.
It has been shown by M. PoisSON, in his first Memoir on Magnetism (Mem. de i'Acad. de Sciences, 1821 et 1822), that when an electrified body is placed in the interior of a hollow spherical conducting shell of uniform thickness, it will not be acted upon in the slightest degree by any bodies exterior to the shell, however intensely they may be electrified. In the ninth article of the present Essay this is proved to be generally true, whatever may be the form or thickness of the conducting shell.
In the tenth article there will be found some simple equations, by means of which the density of the electricity induced on a spherical conducting surface, placed under the influence of any electrical forces whatever, is immediately given; and thence the general value of the potential function for any point either within or without this surface is determined from the arbitrary value at the surface itself, by the aid of a definite integral. The proportion in which the electricity will divide itself between two insulated conducting spheres of different diameters, connected by a very fine wire, is afterwards considered; and it is proved, that when the radius of one of them is small compared with the distance between their surfaces, the product of the mean density of the electricity on either sphere, by the radius of that sphere, and again by the shortest distance of its surface from the centre of the other sphere, will be the same for both. Hence when their distance is very great, the densities are in the inverse ratio of the radii of the spheres.
When any hollow conducting shell is charged with eleccricity, the whole of tbc fluid is carried to the exterior surface, without leaving any p rtion on the interior one, as may he immediately shown from the fourth and fifth articles. In the experimental verification of this, it is necessary to leave a small orifice in the shell: it became therefore a problem of some interest to determine the modification which this alteration would produce. We have, on this account, terminated the present article, by investigating the law of the distribution of electricity on a thin spherical conducting shell, having a small circular orifice, and have found that its density is very nearly constant on the exterior surface, except in the immediate vicinity of the orifice; and the density at any point p of the inner surface, is to the constant density on the outer one, as the product of the diameter of a circle into the cube of the radius of the orifice, is to the product of three times the circumference of that circle into the cube of the distance of p from the centre of the orifice; excepting as before those points in its immediate vicinity. Hence, if the diameter of the sphere were twelve inches, and that of the orifice one inch, the density at the point on the inner surface opposite the centre of the orifice, would be less than the hundred and thirty thousandth part of the constant density on the exterior surface.
In the eleventh article some of the effects due to atmospherical electricity are considered; the subject is not however insisted upon, as the great variability of the cause which produces them, and the impossibility of measuring it, gives a degree of vagueness to these determinations.
The form of a conducting body being given, it is in general a problem of great difficulty, to determine the law of the distribution of the electric fluid on its surface: but it is possible to give different forms, of almost every imaginable variety of shape, to conducting bodies; such, that the values of the density of the electricity on their surfaces may be rigorously assignable by the most simple calculations: the manner of doing this is explained in the twelfth article, and two examples of its use are given. In the last, the resulting form of the conducting body is an oblong spheroid, and the density of the electricity on its surface, here found, agrees with the one long since deduced fronother methods.
Thus far perfect conductors only have been considered. In order to give an example of the application of theory to bodies which are not so, we have, in the thirteenth article, supposed the matter of which they are formed to be endowed with a constant coercive force equal to /9, and analogous to friction in its operation, so that when the resultant of the electric forces acting upon any one of their elements is less than /3, the electrical state of this element shall remain unchanged; but, so soon as it begins to exceed y9, a change shall ensue. Then imagining a solid of revolution to turn continually about its axis, and to be subject to a constant electrical force / acting in parallel right lines, we determine the permanent electrical state at which the body will ultimately arrive. The result of the analysis is, that in consequence of the coercive force fi, the solid will receive a new polarity, equal to that which would be induced in it if it were a perfect conductor and acted upon by the constant force /8, directed in lines parallel to one in the body's equator, making the angle 90° + 7, with a plane passing through its axis and parallel to the direction of/: f being supposed resolved into two forces, one in the direction of the body's axis, the other b directed along the intersection of its equator with the plane just mentioned, and 7 being determined by the equation
sm7 = ^ .
In the latter part of the present article the same problem is considered under a more general point of view, and treated by a different analysis: the body's progress from the initial, towards that permanent state it was the object of the former part to determine is exhibited, and the great rapidity of this progress made evident by an example.
The phenomena which present themselves during the rotation of iron bodies, subject to the influence of the earth's magnetism, having lately engaged the attention of experimental philosophers, we have been induced to dwell a little on the solution of the preceding problem, since it may serve in some measure to illustrate what takes place in these cases. Indeed,