510. Let S be any closed surface, and let O be a point, either external or internal, where a mass, m, of matter is collected. Let N be the component of the attraction of m in the direction of the normal drawn inwards from any point P, of S. Then, if do denotes an element of S, and ff integration over the whole of it,

SS N dσ = 4πm, or = 0,

according as O is internal or external.

Case I., O internal. Let OPP,P,... be a straight line drawn in any direction from O, cutting S in P,, P., P., etc., and therefore passing out at P,, in at P,, out again at P, in again at P, and so

on.

Let a conical surface be described by lines through O, all infinitely near OP,P,..., and let w be its solid angle, (§ 482). The portions of ff Ndo corresponding to the elements cut from S by this cone will be clearly each equal in absolute magnitude to wm, but will be alternately positive and negative. Hence as there is an odd number of them, their sum is + wm. And the sum of these, for all solid angles round O, is (§ 483) equal to 4mm; that is to say, SS Ndo = 4mm.

Case II., O external. Let OPP,P... be a line drawn from O pass ing across S, inwards at P,, outwards at P, and so on. Drawing, as before, a conical surface of infinitely small solid angle, w, we have still wm for the absolute value of each of the portions of ff Ndo corresponding to the elements which it cuts from S; but their signs are alternately negative and positive; and therefore as their number is even, their sum is zero. Hence

SJ Ndo=0.

From these results it follows immediately that if there be any continuous distribution of matter, partly within and partly without a closed surface S, and N and do be still used with the same 'signification, we have

S§Ndo = 4′′ M

if M denote the whole amount of matter within S.

511. From this it follows that the potential cannot have a maximum or minimum value at a point in free space. For if it were so, a closed surface could be described about the point, and indefinitely near it, so that at every point of it the value of the potential would be less than, or greater than, that at the point; so that N would be negative or positive all over the surface, and therefore ƒƒNdo would be finite, which is impossible, as the surface contains none of the attracting mass.

512. It is also evident that I must have positive values at some parts of this surface, and negative values at others, unless it is zero all over it. Hence in free space the potential, if not constant round any point, increases in some directions from it, and diminishes in

Vol. 23-9

others; and therefore a material particle placed at a point of zero force under the action of any attracting bodies, and free from all constraint, is in unstable equilibrium, a result due to Earnshaw'.

513. If the potential be constant over a closed surface which contains none of the attracting mass, it has the same constant value throughout the interior. For if not, it must have a maximum or minimum value somewhere within, which is impossible.

514. The mean potential over any spherical surface, due to matter entirely without it, is equal to the potential at its centre; a theorem apparently first given by Gauss. See also Cambridge Mathematical Journal, Feb. 1845 (vol. iv. p. 225). This proposition is merely an extension, to any masses, of the converse of the following statement, which is easily seen to follow from the results of S$ 479, 488 expressed in potentials instead of forces. The potential of an uniform spherical shell at an external point is the same as if its mass were condensed at the centre. At all internal points it has the same value as at the surface.

515. If the potential of any masses has a constant value, V, through any finite portion, K, of space, unoccupied by matter, it is equal to through every part of space which can be reached in any way without passing through any of those masses: a very remarkable proposition, due to Gauss. For, if the potential differ from Vin space contiguous to K, it must (§ 513) be greater in some parts and less in others.

From any point C within K, as centre, in the neighbourhood of a place where the potential is greater than V; describe a spherical surface not large enough to contain any part of any of the attracting masses, nor to include any of the space external to K except such as has potential greater than V. But this is impossible, since we have just seen (§ 514) that the mean potential over the spherical surface must be . Hence the supposition that the potential is greater than V in some places and less in others, contiguous to K and not including masses, is false.

516. Similarly we see that in any case of symmetry round an axis, if the potential is constant through a certain finite distance, however short, along the axis, it is constant throughout the whole space that can be reached from this portion of the axis, without crossing any of the masses..

517. Let S be any finite portion of a surface, or complete closed surface, or infinite surface, and let E be any point on S. (a) It is possible to distribute matter over S so as to produce potential equal to F (E), any arbitrary function of the position of E, over the whole of S. (b) There is only one whole quantity of matter, and one distribution of it, which can satisfy this condition. For the proof of

1 Cambridge Phil. Trans., March, 1839.

this and of several succeeding theorems, we refer the reader to our larger work.

518. It is important to remark that, if S consist, in part, of a closed surface, Q, the determination of U, the potential at any point, within it will be independent of those portions of S, if any, which lie without it; and, vice versa, the determination of U through external space will be independent of those portions of S, if any, which lie within the part Q. Or if S consist, in part, of a surface Q, extending infinitely in all directions, the determination of U through all space on either side of Q, is independent of those portions of S, if any, which lie on the other side.

519. Another remark of extreme importance is this:-If F(E) be the potential at E of any distribution, M, of matter, and if S be such as to separate perfectly any portion or portions of space, H, from all of this matter; that is to say, such that it is impossible to pass into H from any part of M without crossing S; then, throughout H, the value of U will be the potential of M.

520. Thus, for instance, if S consist of three detached surfaces, S, S., S, as in the diagram, of which S, S, are closed, and S. is an open shell, and if F (E) be the potential due to M, at any point, E, of any of these portions of S; then throughout H, and H,, the spaces within S, and without S, the value of U is simply the potential of M. The value of U through K, the remainder of space, depends, of course, on the character of the composite surface S.

K

H, H.

K

521. From § 518 follows the grand proposition:-It is possible to find one, but no other than one, distribution of matter over a surface S which shall produce over S, and throughout all space H separated by S from every part of M, the same potential as any given mass M.

Thus, in the preceding diagram, it is possible to find one, and but one, distribution of matter over S, S., S, which shall produce over S, and through H, and H, the same potential as M.

The statement of this proposition most commonly made is: It is possible to distribute matter over any surface, S, completely enclosing a mass M, so as to produce the same potential as M through all space outside M; which, though seemingly more limited, is, when interpreted with proper mathematical comprehensiveness, equivalent to the foregoing.

522. If S consist of several closed or infinite surfaces, S1, S., S2, respectively separating certain isolated spaces H1, H., H,, from H, the remainder of all space, and if F (E) be the potential of masses m1, m,, m,, lying in the spaces H,, H, H; the portions of U due to

S, S., S., respectively will throughout H be equal respectively to the potentials of m1, m,, m,, separately.

For, as we have just seen, it is possible to find one, but only one, distribution of matter over S, which shall produce the potential of m1,

H

H

H

throughout all the space H, H., H,, etc., and one, but only one, distribution over S, which shall produce the potential of me throughout H, H, H, etc.; and so on. But these distributions on S, S,, etc., jointly constitute a distribution producing the potential F(E) over every part of S, and therefore the sum of the potentials due to them all, at any point, fulfils the conditions presented for U. This is therefore (§ 518) the solution of the problem.

H

H

523. Considering still the case in which F (E) is prescribed to be the potential of a given mass, M: let S be an equipotential surface enclosing M, or a group of isolated surfaces enclosing all the parts of M, and each equipotential for the whole of M. The potential due to the supposed distribution over S will be the same as that of M, through all external space, and will be constant (§ 514) through each enclosed portion of space. Its resultant attraction will therefore be the same as that of M on all external points, and zero on all internal points. Hence we see at once that the density of the matter distributed over it, to produce F (E), is equal to where R denotes the resultant force of M, at the point E.

R

4π

524. When M consists of two portions m, and m' separated by an equipotential S, and S consists of two portions, S, and S', of which the latter separates the former perfectly from m'; we see, by $ 522, that the distribution over S, produces through all space on the side of it on which S lies, the same potential, V, as m, and the distribution on S produces through space on the side of it on which S, lies, the same potential, V, as m'. "But the supposed distribution on the whole of S'is such as to produce a constant potential, C, over S1,

and consequently the same at every point within S. Hence the in ternal potential, due to S, alone, is C1-V'.

Thus, passing from potentials to attractions, we see that the resultant attraction of S, alone, on all points on one side of it, is the same as that of m1; and on the other side is equal and opposite to that of the remainder m' of the whole mass. The most direct and simple complete statement of this result is as follows:

If masses m, m', in portions of space, H, H', completely separated from one another by one continuous surface S, whether closed or infinite, are known to produce tangential forces equal and in the same direction at each point of S, one and the same distribution of matter over S will produce the force of m throughout H', and that of m' throughout H. The density of this distribution is equal to R if R denote the resultant force due to one of the masses, and the other with its sign changed. And it is to be remarked that the direction of this resultant force is, at every point, E, of S, perpendicular to S, since the potential due to one mass, and the other with its sign changed, is constant over the whole of S.

4π

525. Green, in first publishing his discovery of the result stated in § 523, remarked that it shows a way to find an infinite variety of closed surfaces for any one of which we can solve the problem of determining the distribution of matter over it which shall produce a given uniform potential at each point of its surface, and consequently the same also throughout its interior. Thus, an example which Green himself gives, let M be a uniform bar of matter, AA'. The equipotential surfaces round it are, as we have seen above ($ 499 (e)), prolate ellipsoids of revolution, each having A and A' for its foci; and the resultant force at C was found to be

the whole mass of the bar being denoted by m, its length by 2a, and A'C+AC by 27. We conclude that a distribution of matter over the surface of the ellipsoid, having

I m.CF 4π l (l3 — a3)

for density at C, produces on all external space the same resultant force as the bar, and zero force or a constant potential through the internal space. This is a particular case of the general result regarding ellipsoidal shells, proved below, in §§ 536, 537.

526. As a second example, let M consist of two equal particles, at points I, I'. If we take the mass of each as unity, the potential af