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which values are correct to quantities of the order, or, which is the same thing, to quantities of the order 0; these having been neglected in the latter part of the preceding analysis, as unworthy of notice.

1

Suppose do is an element of the surface A, the corresponding element of B, cut off by normals to 4, will be do 1+0+), and therefore the quantity of fluid on this last element will be odo {1+0(1+1)}; sub

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R

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the same quantity as on the element do of the first surface. If therefore, we conceive any portion of the surface A, bounded by a closed curve, and a corresponding portion of the surface B, which would be cut off by a normal. to A, passing completely round this curve; the sum of the two quantities of electric fluid, on these corresponding portions, will be equal to zero; and consequently, in an electrical jar any how charged, the total quantity of electricity in the jar may be found, by calculating the quantity, on the two exterior surfaces of the metallic coatings farthest from the glass, as the portions of electricity, on the two surfaces adjacent to the glass, exactly neutralise each other. This results will appear singular, when we consider the immense quantity of fluid collected on these last surfaces, and moreover, it would not be difficult to verify it by experiment.

As a particular example of the use of this general theory: suppose a spherical conductor whose radius a, to communicate with the inside of an electrical jar, by means of a long slender wire, the outside being in communication with the common reservoir; and let the whole be charged: then P representing the density of the electricity on the surface of the conductor, which will be very nearly constant, the value of the potential function within the sphere, and, in consequence of the communication established, at the inner coating A also, will be 4лaP very nearly, since we may, without sensible

error, neglect the action of the wire and jar itfelf in calculating it. Hence

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and the equations (8), by neglecting quantities of the order 0, give

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We thus obtain, by the most simple calculation, the values of the densities, at any point on either of the surfaces A and B, next the glass, when that on the spherical conductor is known.

The theory of the condenser, electrophorous, etc. depends upon what has been proved in this article; but these are details into which the limits of this Essay will not permit me to enter; there is, however, one result, relative to charging a number of jars by cascade, that appears worthy of notice, and which flows so readily from the equations (8), that I cannot refrain from introducing it here.

Conceive any number of equal and similar insulated Leyden phials, of uniform thickness, so disposed, that the exterior coating of the first, may communicate with the interior one of the second; the exterior one of the second, with the interior one of the third; and so on throughout the whole series, to the exterior surface of the last, which we will suppose in communication with the earth. Then, if the interior of the first phial, be made to communicate with the prime conductor of an electrical machine, in a state of action, all the phials will receive a certain charge, and this mode of operating is called charging by cascade. Permitting ourselves to neglect the small quantities of free fluid on the exterior surfaces of the metallic coatings, and other quantities of the same order, we may readily determine the electrical state of each phial in the series: for thus, the equations (8) become

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Designating now, by an index at the foot of any letter, the number of the phial to which it belongs, so that, o, may belong to the first, 2 to the second phial, and so on; we shall have, by supposing their whole number to be n, since is the same for every one,

Crelle's Journal f. d. M. Bd. XLVII. Heft 2.

22

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Now represents the value of the total potential function, within the prime conductor and interior coating of the first phial, and in consequence of the communications established in this system, we have in regular succession, beginning with the prime conductor, and ending with the exterior surface of the last phial, which communicates with the earth,

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=

But the first system of equations gives 0=2+, whatever whole number s may be, and the second line of that just exhibited is expressed by 0=0,-1+0,, hence by comparing these two last equations

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which shows that every phial of the system is equally charged. Moreover, if we sum up vertically, each of the columns of the first system, there will arise in virtue of the second

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We therefore see, that the total charge of all the phials is precisely the same, as that which one only would receive, if placed in communication with the same conductor, provided its exterior coating were connected with the earth. Hence this mode of charging, although it may save time, will never produce a greater accumulation of fluid, than would take place, if one phial only were employed.

9.

Conceive now, a hollow shell of perfectly conducting matter, of any form and thickness whatever, to be acted upon by any electrified bodies, situate without it; and suppose them to induce an electrical state in the shell;

then will this induced state be such, that the total action on an electrified particle, placed any where within it, will be absolutely null.

For let V represent the value of the total potential function, at any point p within the shell, then we shall have at its inner surface, which is a closed one,

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B being the constant quantity, which expresses the value of the potential function, within the substance of the shell, where the electricity is, by the supposition, in equilibrium, in virtue of the actions of the exterior bodies, combined with that arising from the electricity induced in the shell itself. Moreover, V evidently satisfies the equation 0=SV, and has no singular value within the closed surface to which it belongs: it follows therefore, from art. 5, that its general value is

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and as the forces acting upon p, are given by the differentials of V, these forces are evidently all equal to zero.

If, on the contrary, the electrified bodies are all within the shell, and its exterior surface is put in communication with the earth, it is equally easy to prove, that there will not be the slightest action on any electrified point exterior to it; but, the action of the electricity induced on its inner surface, by the electrified bodies within it, will exactly balance the direct action of the bodies themselves. Or more generally:

Suppose we have a hollow, and perfectly conducting shell, bounded by any two closed surfaces, and a number of electrical bodies are placed, some within and some without it, at will; then, if the inner surface and interior bodies be called the interior system; also, the outer surface and exterior bodies the exterior system; all the electrical phenomena of the interior system, relative to attractions, repulsions, and densities, will be the same as would take place if there were no exterior system, and the inner surface were a perfect conductor, put in communication with the earth; and all those of the exterior system will be the same, as if the interior one did not exist, and the outer surface were a perfect conductor, containing a quantity of electricity, equal to the whole of that originally contained in the shell itself, and in all the interior bodies.

This is so direct a consequence of what has been shown in articles 4 and 5, that a formal demonstration would be quite superfluous, as it is easy to see, the only difference which could exist, relative to the interior system,

between the case where there is an exterior system, and where there is not one, would be in the addition of a constant quantity, to the total potential function within the exterior surface, which constant quantity must necessarily disappear in the differentials of this function, and consequently, in the values of the attractions, repulsions, and densities, which all depend on these differentials alone. In the exterior system there is not even this difference, but the total potential function exterior to the inner surface is precisely the same, whether we suppose the interior system to exist or not.

10.

The consideration of the electrical phenomena, which arise from spheres variously arranged, is rather interesting, on account of the case with which all the results obtained from theory, may be put to the test of experiment; but, the complete solution of the simple case of two spheres only, previously electrified, and put in presence of each other, requires the aid of a profound analysis, and has been most ably treated by M. Poisson (Mém. de l'Institut. 1811). Our object, in the present article, is merely to give one or two examples of determinations, relative to the distribution of electricity on spheres, which may be expressed by very simple formulae.

Suppose a spherical surface whose radius is a, to be covered with electric matter, and let its variable density be represented by o; then if, as in the Méc. Céleste, we expand the potential function V, belonging to a point p within the sphere, in the form

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r being the distance between p and the centre of the sphere, and U, U, etc. functions of the two other polar co-ordinates of p, it is clear, by what has been shown in the admirable work just mentioned, that the potential function V', arising from the same spherical surface, and belonging to a point p', exterior to this surface, at the distance r' from its centre, and on the radius produced, will be

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If, therefore, we make V=4(r), and V'=y(r'), the two functions φ and Ψ will satisfy the equation

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