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3u> ~d~r dr dr* o*'
becomes — 4ir V
when the radius a is supposed to vanish. Thus, the equation (2) becomes
jdxdydzUBV+jda Uj£~ jdxdydz VB U+jda V-^-torV'... (3);
where, as in the former equation, the triple integrals extend over the whole volume of the body, and those relative to da, over its exterior surface: V being the value of Fat the point p.
In like manner, if the function V be such, that it becomes infinite for any point p" within the body, and iB moreover,
sensibly equal to -,, infinitely near this point, as 27 is infinitely
near to the point p', it is evident from what has preceded that we shall have
Idxdydz UB V+jda 0^-4t #"=\dxdydz VB U+fda V^-iirV...^');
the integrals being taken as before, and U" representing the value of U, at the point p" where V becomes infinite. The same process will evidently apply, however great may be the number of similar points belonging to the functions Z7and V.
For abridgment, we shall in what follows, call those singular values of a given function, where its differential coefficients become infinite, and the condition originally imposed upon U and V will be expressed by saying, that neither of them has any singular values within the solid body under consideration.
(4.) We will now proceed to determine some relations existing between the density of the electric fluid at the surface of a body, and the potential functions thence arising, within and without this surface. For thiB, let pda be the quantity of electricity on an element da of the surface, and V, the value of the potential function for any point p within it, of which the co-ordinates are x, y, s. Then, if V be the value of this function for any other pointy exterior to this surface, we shall have
th* Integrals relative to do- extending over the whole surface of tlw Ikhiy.
It might appear at first view, that to obtain the value of V fam that of V, we should merely have to change x, y, z into tt\ y*, #': but, this is by no means the case; for, the form of the potential function changes suddenly, in passing from the space within to that without the surface. Of this, we may give a very «iui|ilo example, by supposing the surface to be a sphere whose twliu* in a and centre at the origin of the co-ordinates; then, if the density p be constant, we shall have
F-4*pa and F—-= 4TM--^Y
which are essentially distinct functions.
With respect to the functions Fand V in the general case, it 1m clear that each of them will satisfy Laplace's equation, and eotvsefjucntly
0 = SFand 0 = 8T':
inure over, neither of them will have singular values; for any point of the spaces to which they respectively belong, and at the miiTaoc itself, we shall have
dm horizontal lines over the quantities indicating that they belong to the surface. At an infinite distance from this surface, we shall likewise have
F* = 0.
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 thera 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 £7"= -, has but one singular point, viz. p; and, we have
also 8 V= 0 and S - = 0: r beiug the distance between the point
p to which V belongs, and the element da:
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 hero
-*=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 point p' 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, - (-*- J = force 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 — (-r-r) 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 2wp, 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 V and V would still be given by the formulae
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 tins 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
dV dV which could do so: p, -j— and -j-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
Q = 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,
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 wonld become