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 $$ 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 V. Hence the supposition that the potential is greater than 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, S1, 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 H1 and H2, 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 S1, 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, S2, S3, respectively separating certain isolated spaces H1, H2, H2, from H, the remainder of all space, and if F (E) be the potential of masses mi, m,, m,, lying in the spaces H1, H, H; the portions of U due to

S1, S2, S1, respectively will throughout H be equal respectively to the potentials of m1, m,, m2, 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

H
H

H1

throughout all the space H, H2, H2, etc., and one, but only one, distribution over S, which shall produce the potential of ma throughout H, H1, H2, 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.

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 dis

R

4π

tributed over it, to produce F (E), is equal to where R denotes the resultant force of M, at the point E.

524. When M consists of two portions m, and m' separated by an equipotential S1, 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, V1, as m1, 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, C1, over S1,

and consequently the same at every point within S. Hence the internal potential, due to S, alone, is C-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

4π

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.

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 2l. We conclude that a distribution of matter over the surface of the ellipsoid, having

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 at

=

C

is the equation of an equipotential surface; it being understood that negative values of IP and I'P are inadmissible, and that any constant value, from ∞ to o, may be given to C. The curves in the annexed diagram have been drawn, from this equation, for the cases of C equal respectively to 10, 9, 8, 7, 6, 5, 45, 43, 4'2, 41, 4, 3'9, 38, 37, 35, 3, 25, 2; the value of II' being unity.

The corresponding equipotential surfaces are the surfaces traced by these curves, if the whole diagram is made to rotate round II' as

P

axis. Thus we see that for any values of C less than 4 the equipotential surface is one closed surface. Choosing any one of these surfaces, let R denote the resultant of forces equal top and p

I

I

Then if matter be distributed over this
R

in the lines PI and PI'.
surface, with density at P equal to
"
Απ ́

its attraction on any internal point will be zero; and on any external point, will be the same as that of I and I'.

527. For each value of C greater than 4, the equipotential surface consists of two detached ovals approximating (the last three or four in the diagram, very closely) to spherical surfaces, with centres lying between the points I and I', but approximating more and more closely to these points, for larger and larger values of C.

Considering one of these ovals alone, one of the series enclosing I', for instance, and distributing matter over it according to the same law of density, we have a shell of matter which exerts (§ 525)

R

4π