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We will now show, that if any two functions whatever are taken, satisfying these conditions, it will always be in our power to assign one, and only one value of p, which will produce them for corresponding potential functions. For this we may remark, that the Equation (3) art. 3 being applied to the space within
the body, becomes, by making U— - ,
since TJ*=- -, has but one singular point, viz. p; and, we have also 8 F*= 0 and S - = 0: r being the distance between the point
p to which V belongs, and the element dir.
If now, we conceive a surface inclosing the body at an infinite distance from it, we shall have, by applying the formula (2) of the same article to the space between the surface of the body and this imaginary exterior surface (seeing that here
~. = U has no singular value)
since the part due to the infinite surface may be neglected, because V is there equal to zero. In this last equation, it is evident that dw' is measured from the surface, into the exterior space, and hence
which equation reduces the sum of the two just given to
In exactly the same way, for the pointy' exterior to the surface, we shall obtain
Hence it appears, that there exists a value of p, viz.
which will give Fand V, for the two potential functions, within and without the surface.
Again, — (gj~J" f°rce with which a particle of positive
electricity p, placed within the surface and infinitely near it, is impelled in the direction dw perpendicular to this surface,
and directed inwards; and — j expresses the force with
which a similar particle p placed without this surface, on the same normal with p, and also infinitely near it, is impelled outwards in the direction of this normal: but the sum of these two forces is equal to double the force that an infinite plane would exert upon p, supposing it uniformly covered with electricity of the same density as at the foot of the normal on which p is; and this last force is easily shown to be expressed by 2irp, hence by equating
and consequently there is only one value of p, which can produce V and V as corresponding potential functions.
Although in what precedes, we have considered the surface of one body only, the same arguments apply, how great soever may be their number; for the potential functions Fand V would still be given by the formula}
the only difference would be, that the integrations must now extend over the surface of all the bodies, and, that the number of functions represented by V, would be equal to the number of the bodies, one for each. In this case, if there were given a value of V for each body, together with V belonging to the exterior space; and moreover, if these functions satisfied to the above mentioned conditions, it would always be possible to determine the density on the surface of each body, so as to produce these values as potential functions, and there would be but one density, viz. that given by
which could do so: p, ^ and —r belonging to a point on the surface of any of these bodies.
(5.) From what has been before established (art. 3), it is easy to prove, that when the value of the potential function V is given on any closed surface, there is but one function which can satisfy at the same time the equation
0 = BV,
and the condition, that V shall have no singular values within this surface. For the equation (3) art 3, becomes by supposing BU=0,
J aw j aw
In this equation, U is supposed to have only one singular value within the surface, viz. at the point p', and, infinitely near to
this point, to be sensibly equal to -; r being the distance
from p'. If now we had a value of U, which, besides satisfying the above written conditions, was equal to zero at the surface itself, we should have Z7= 0, and this equation would become
which shows, that V the value of V at the point p is given, when V its value at the surface is known.
To convince ourselves that there does exist such a function as we have supposed U to be; conceive the surface to be a perfect conductor put in communication with the earth, and a unit of positive electricity to be concentrated in the point p', then the total potential function arising from p and from the electricity it will induce upon the surface, will be the required value of IT. For, in consequence of the communication established between the conducting surface and the earth, the total potential function at this surface must be constant, and equal to that of the earth itself, i. e. to zero (seeing that in this state they form but one conducting body). Taking, therefore, this total potential function for U, we have evidently 0=17, 0 = 817, and C= - for
those parts infinitely near to p. As moreover, this function has no other singular points within the surface, it evidently possesses all the properties assigned to U in the preceding proof.
Again, since we have evidently V = 0, for all the space exterior to the surface, the equation (4) art. 4 gives
where (p) is the density of the electricity induced on the surface, by the action of a unit of electricity concentrated in the point p. Thus, the equation (5) of this article becomes
This equation is remarkable on account of its simplicity and singularity, seeing that it gives the value of the potential for any point p, within the surface, when V, its value at the surface itself is known, together with (p), the density that a unit of electricity concentrated in p would induce on this surface, if it conducted electricity perfectly, and were put in communication with the earth.
Having thus proved, that V the value of the potential function V, at any point p within the surface is given, provided its value V is known at this surface, we will now show, that whatever the value "of V may be, the general value of V deduced from it by the formula just given shall satisfy the equation
0 = 87.
For, the value of V at any point p whose co-ordinates are x, y, z, deduced from the assumed value of V, by the above written formula, is
U being the total potential function within the surface, arising from a unit of electricity concentrated in the point p, and the electricity induced on the surface itself by its action. Then, since V is evidently independent of x, y, z, we immediately deduce
Now the general value of U will depend upon the position of the point p producing it, and upon that of any other point p whose co-ordinates are x, y', z, to which it is referred, and will consequently be a function of the six quantities, x, y, z, x, y', z'.
But we may conceive U to be divided into two parts, one *»
(r being the distance pp') arising from the electricity in p, the other, due to the electricity induced on the surface by the action of p, and which we shall call U4. Then since U, has no singular values within the surface, we may deduce its general value from that at the surface, by a formula similar to the one just given. Thus
where V is the total potential function, which would be produced by a unit of electricity in^', and therefore, (^^j ia impendent of the co-ordinates x, y, z, of j», to which S refers.