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Take any continuous function V, of the rectangular coordinates x', y, z, of a point p', which satisfies the partial differential equation 0 = BY', and vanishes when p is removed to an infinite distance from the origin of the co-ordinates.

Choose a constant quantity b, such that V'=b may he the equation of a closed surface A, and that V may have no singular values, so long as p is exterior to this surface: then if we form a conducting body, whose outer surface is A, the density of the electric fluid in equilibrium upon it, will be represented by

= -h <W_ P ~4tt dw'

and the potential function due to this fluid, for any point p, exterior to the body, will be

hV;

h being a constant quantity dependent upon the total quantity of electricity Q, communicated to the body. This is evident from what has been proved in the articles cited.

Let R represent the distance between p, and any point within A; then the potential function arising from the electricity upon it will be expressed by ^, when R is infinite. Hence the condition

j^ = hV (R being infinite),

which will serve to determine k, when Q is given.

In the application of this general method, we may assume for V, either some analytical expression containing the coordinates of p, which is known to satisfy the equation 0 = BY', and to vanish when p' is removed to an infinite distance from the origin of the co-ordinates; as, for instance, some of those given by Laplace (Mec. Celeste, Liv. 3, Ch. 2), or, the value fr* a potential function, which would arise from a quantity of electricity anyhow distributed within a finite space, at a point p without that space; since this last will always satisfy the conditions to which V is subject.

It may be proper to give an example of each of these cases. In the first place, let ns take the general expression given by Laplace,

tfPl 77(il rjm

then, by confining ourselves to the two first terms, the assumed value of V will be

'_'„!,

T T

r being the distance of p from the origin of the- co-ordinates, and Um, Um, &c. functions of the two other polavco-ordinates 6 and vr. This expression by changing the direction of the axes, may always be reduced to the form

y, _ 2a cos 0
--7 + —p—i

a and Je being two constant quantities, which we^ will suppose positive. Then if b be a very small positive quantity, the form of the surface given by the equation V' b, will differ but little

2a

from a sphere, whose radius is :by gradually increasing b,

the difference becomes greater, until 5 = ^; and afterwards, the

form assigned by 7=5, becomes improper for our purpose.

Making therefore b = p, in order to have a surface differing as

much from a sphere, as the assumed value of V admits, the equation of the surface A becomes

„, _ 2a k* cos 0 _ a*

V ~V+ r* ~¥'
From which we obtain

ifc* / 0\
r = - l + V2cos- .
a \ 2/

If now $ represents the angle formed by dr and aV, we have

and as the electricity is in equilibrium upon A, the force with •which a particle p, infinitely near to it, would be repelled, must

dV'

be directed along dw': but the value of this force is — -y-j, and

consequently its effect in the direction of the radius r, and tend

dV'

ing to increase it, will be — ^-cos^. This last quantity is

dV'

equally represented by — -^r-, and therefore dV' dV' .

—iT—dirTM*,

the horizontal lines over quantities, indicating, as before, that they belong to the surface itself. The value of — f^rj , deduced from this equation, is

dV'^ -1 dV'm 1 (2a i 2l£ cos 6\ m 2°^2 008 2
dw cos dr, cos^jr" V J r" cos <f>'

this substituted in the general value of p, before given, there arises

_~hdT Ao^2cos|

4w dw' 2irrt cos '

Supposing Q is the quantity of electricity communicated to the surface, the condition

^ = hV (where B is infinite),

before given, becomes, since r may here be substituted for R, seeing that it is measured from a point within the surface,

Q 2ah . Q

— = i.e. n = —.

r r 2a

We have thus the rigorous value of p for the surface A whose

equation is r = ^1 + ^2 cos ^ when the quantity Q of elec

tricity upon it is known, and by substituting for r and h their values just given, there results

g

Qc?Y2 cos 4*^008 ^1 + </2 cos^'

Moreover the value of the potential function for the point p whose polar co-ordinates are r, 0, and «r, is

From which we may immediately deduce the forces acting on any pointy' exterior to A.

In tracing the surface A, 8 is supposed to extend from 0 = 0 to 0 = ir, and «r, from w = 0 to « = 2tt: it is therefore evident, by constructing the curve whose equation is

that the parts about P, where 8 = ir, approximate continually in form towards a cone whose apex is. P, and as the density of the electricity at P is null, in the example before us, we may make this general inference: when any body whatever has a part of its surface in the form of a cone, directed inwards; the density of the electricity in equilibrium upon it, will be null at its apex, precisely the reverse of what would take place, if it were directed outwards, for then, the density at the apex would become infinite*.

'Since this was written, I have obtained formula; serving to express, generally, the law of the distribution of the electric fluid near the apex 0 of a cone, which forms part of a conducting surface of revolution having the same axis. From these formulae it results that, when the apex of the cone is directed inwards, the density of the electric fluid at any point p, near to it, is proportional to r*-1; r being the distance Op, and the exponent » very nearly such as would satisfy the simple equation (411 + 1) p— 3*-: where »/3 is the angle at the summit of the cone. H 1J3 exceeds r, this summit is directed outwards, and when the excess is sot very considerable, n will be given as above: but »/3 still increasing, until it becomes it - 2y the angle ty at the summit of the cone, which is now directed

outwards, being very small, n will be given by an log ^= 1, and in case the con

Auctiog body is * sphere whose radios is b, on which P represents the mean density

As a second example, we will assume for V, the value of the potential function arising from the action of a line uniformly covered with electricity. Let 2a be the length of the line, y the perpendicular falling from any point p' upon it, x the distance of the foot of this perpendicular from the middle of the line, and x' that of the element dx' from the same point: then taking the element dx', as the measure of the quantity of electricity it contains, the assumed value of V will be

yx [ M. W «-tt + vV+(«-g)'}.

) V W + (* - *')* i ~ g - « - * + VV + (« + xf)' the integral being taken from x' = — a to x' = + a. Making this equal to a constant quantity log b, we shall have, for the equation of the surface A,

a-x + *J{y*+(a-xy\ -a-x + JW+ia + x)*}-0'

which by reduction becomes

0 = y' (1 - Vf + x\ ib (1 - by - o\ 4o (1 + b)\

We thus see that this surface is a spheroid produced by the revolution of an ellipsis about its greatest diameter; the semitransverse axis being

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1-b

By differentiating the general value of V, just given, and substituting for y its value at the surface A, we obtain

1-b

2x '. T

dV 1 + 0 -2aftx

of the electric fluid, p, the value of the density near the apex 0, will he determined by the formula

tPbn AY*-1 P = (a+b)y W'

a being the length of the cone.

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