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It now only remaius to determine the value of V from this equation. For this, let j8 now represent the linear radius of *, and 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 JPy arising from this spheroid, will be
T/ being the distance do; C. and k a constant quantity. The attraction exerted by this spheroid, in the direction of the perpendicular a, will be — ~, and by the known formulae relative
to the attractions of homogeneous spheroids, we have d<f> SMz . .
M representing the mass of the spheroid, and 6 being determined by the equations
Supposing now z very small, since it is to vanish at the end of the calculus, and y </3, in order that the point P may fall within tht limits of s, we shall have by neglecting quantities of the order 2 compared with those retained
This expression, being differentiated again relative to z, gives <** 1 (da t/£H ~ ZMir
which expression is rigorously exact when z — 0. Comparing this result with the equation (12) of the present article, we see that if V=k^(p* — if), the constant quantity k may be always determined, so as to satisfy (12). In fact, we have only to make
— Fir - F
•tfk = i. e. k = .
Having thus the value of V, the general value of V is known, since
as-5' dd> a*-a» UTz u . .
= -,-— x - -J- = x „, (tan 6-6)
iiraz az iiraz p 1'
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
47r dw 4tt db 4w*a
in which equation, the point P is supposed to be upon the element do1 of the interior surface, to which p belongs. If now R be the distance between 0, the centre of the orifice, and da,
we shall have IP = y* + zi, and by neglecting quantities of the /3s
order ^ compared with those retained, we have successively a~B, 0 = |, and tan*-0 = $<P»jj.
Thus the value of p becomes
, F_ £
9 ~ 12ir8a ,5s 1
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 da" of the exterior surface of the shell, corresponding to the element da of the interior surface, will be
» F ^ < P =4^ +''
which, on account of the smallness of p for every part of the surface, except very near the orifice s, is sensibly constant and F
equal to -—, therefore
P .... P .
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 dxdydz 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 eves a passing cloud,) and r being the distance of this element' from A'b centre; also r its distance from B's, the value of the potential function at A's centre, arising from the whole exterior space, will be
f pdxdydz j—7—.
and the value of the same function at Bya centre will be
the integrals extending over all the space exterior to the conducting system under consideration.
If now, Q be the total quantity of electricity on A'n surface, and Q that on B's, their radii being a and a'; it is clear, the value of the potential function at A'a centre, arising from the system itself, will be
seeing that, we may neglect the part due to the wire, on account of its fineness, and that due to the other sphere, on account of its distance. In a similar way, the value of the same function at i?'s centre will be found to be
But (art. 1) the value of the total potential function must be constant throughout the whole interior of the conducting system, and therefore its value at the two centres must be equal; hence
Q f pdxdydz _ Q' f pdxdydz
Although p, in the present case, is exceedingly small, the integrals contained in this equation may not only be considerable, but very great, since they are of the second dimension relative to space. The spheres, when at a great distance from each other, may therefore become highly electrical, according to the observations of experimental philosophers, and the charge they will receive in any proposed case may readily be calculated; the vulue of p being supposed given. When one of the spheres, B for instance, is connected with the ground, Q will be equal to zero, and consequently Q immediately given. If, on the contrary, the whole system were insulated and retained its natural quantity of electricity, we should have, neglecting that on the wire,
and hence Q and Q would be known.
If it were required to determine the electrical state of the sphere A, when in communication with a wire, of which one extremity is elevated into the atmosphere, and terminates in a fine point p. we should only have to make the radius of B, and consequently, Q', vanish in the expression before given. Hence in this case
Q_ (pdxdydz tpdxdydz
r being the distance between p and the element dxdydz. Since the object of the present article is merely to indicate the cause of some phenomena of atmospherical electricity, it is useless to extend it to a greater length, more particularly as the extreme difficulty of determining correctly the electrical state of the atmosphere at any given time, precludes the possibility of putting this part of the theory to the test of accurate experiment.
(12.) Supposing the form of a conducting body to be given, it is in general impossihle to assign, rigorously, the law of the density of the electric fluid on its surface in a state of equilibrium, when not acted upon by any exterior bodies, and, at present, there has not even been found any convenient mode of approximation applicable to this problem. It is, however, extremely easy to give such forms to conducting bodies, that this law shall be rigorously assignable by the most simple means. The following method, depending upon art. 4 and 5, seems to give to these forms the greatest degree of generality of which they are susceptible, as, by a tentative process, any form whatever might he approximated indefinitely.