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We therefore see, that when the distance b between the centres of the spheres is very great, the mean densities will be inversely as the radii; and these last remaining unchanged, the density on the smaller sphere will decrease, and that on the larger increase in a very simple way, by making them approach each other.

Lastly, let us endeavour to determine the law of the distribution of the electric fluid, when in equilibrium on a very thin spherical shell, in which there is a small circular orifice. Then, if we neglect quantities of the order of the thickness of the shell, compared with its radius, we may consider it as an infinitely thin spherical surface, of which the greater segment S is a perfect conductor, and the smaller one s constitutes the circular orifice. In virtue of the equilibrium, the value of the potential function, on the conducting segment, will be equal to a constant quantity, as F, and if there were no orifice, the corresponding value of the density would be

F
Απα

a being the radius of the spherical surface. Moreover on this supposition, the value of the potential function for any point P,

F

within the surface, would be F. Let therefore, +p re

Απα

present the general value of the density, at any point on the surface of either segment of the sphere, and F+V, that of the corresponding potential function for the point P. The value of the potential function for any point on the surface of the sphere will be F+V, which equated to F, its value on S, gives for the whole of this segment

0 = V.

Thus the equation (10) of this article becomes

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the integral extending over the surface of the smaller segment s only, which, without sensible error, may be considered as a plane.

But, since it is evident that p is the density corresponding to the potential function V, we shall have for any point on the segment 8, treated as a plane,

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as it is easy to see, from what has been before shown (art. 4); dw being perpendicular to the surface, and directed towards the centre of the sphere; the horizontal line always serving to indicate quantities belonging to the surface. When the point P is very near the plane s, and z is a perpendicular from P upons, z will be a very small quantity, of which the square and higher powers may be neglected. Thus ba-z, and by

substitution

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the integral extending over the surface of the small plane s, and f being, as before, the distance P, do. Now

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provided we suppose z = 0 at the end of the calculus. Now the

F

density +p, upon the surface of the orifice s, is equal to

Απα

zero, and therefore we have for the whole of this surface

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the integral extending over the whole of the plane s, of which do is an element, and z being supposed equal to zero, after all the operations have been effected.

It now only remains to determine the value of V from this equation. For this, let 8 now represent the linear radius of s, and y, the distance between its centre C and the foot of the perpendicular z: then if we conceive an infinitely thin oblate spheroid, of uniform density, of which the circular plane s constitutes the equator, the value of the potential function at the point P, arising from this spheroid, will be

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7 being the distance do, C, and k a constant quantity. The attraction exerted by this spheroid, in the direction of the perpendicular z, will be

do
dz'

and by the known formulæ relative

to the attractions of homogeneous spheroids, we have

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M representing the mass of the spheroid, and being determined

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Supposing now z very small, since it is to vanish at the end of the calculus, and y<ß, in order that the point P may fall within the limits of s, we shall have by neglecting quantities of the order a compared with those retained

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This expression, being differentiated again relative to z, gives

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which expression is rigorously exact when z=0. Comparing this result with the equation (12) of the present article, we see that if = k√√(82-n), the constant quantity k may be always determined, so as to satisfy (12). In fact, we have only to make

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Having thus the value of V, the general value of V is known,

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The value of the potential function, for any point P within the shell, being F+V, and that in the interior of the conducting matter of the shell being constant, in virtue of the equilibrium, the value p' of the density, at any point on the inner surface of the shell, will be given immediately by the general formula (4) art. 4. Thus

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in which equation, the point P is supposed to be upon the element do' of the interior surface, to which p' belongs. If now R be the distance between C, the centre of the orifice, and do',

we shall have R2 = y + z2, and by neglecting quantities of the

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In the same way, it is easy to show from the equation (11) of this article, that p", the value of the density on an element do" of the exterior surface of the shell, corresponding to the element do' of the interior surface, will be

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which, on account of the smallness of p' for every part of the surface, except very near the orifice s, is sensibly constant and

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which equation shows how very small the density within the shell is, even when the orifice is considerable.

(11.) The determination of the electrical phenomena, which result from long metallic wires, insulated and suspended in the atmosphere, depends upon the most simple calculations. As an example, let us conceive two spheres A and B, connected by a long slender conducting wire; then pdxdydz representing the quantity of electricity in an element dcdydz of the exterior space, (whether it results from the ground in the vicinity of the wire having become slightly electrical, or from a mist, or even a passing cloud,) and r being the distance of this element from A's centre; also r' its distance from B's, the value of the

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