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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, l, the determination of U within it will be independent of those portions of S, if any, which lie without it; and, vice versá, 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 , 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,, Sy, 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 with
K out Sy, 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.
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, S2, 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, S, S., S3, respectively separating certain isolated spaces H, H2, H3, from H, the remainder of all space, and if F (E) be the potential of masses my, my, mg, lying in the spaces H, H., Hg; the portions of U due to
S, S, Sy, respectively will throughout H be equal respectively to the potentials of my, mq, mg, 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 mi,
throughout all the space H, H,, H, etc., and one, but only one,
distribution over S, which shall H
produce the potential of m, throughout H, H, H, etc.; and
But these distributions
on S, S, etc., jointly constitute H
a distribution producing the poH
tential 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 (8 518) 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 tributed over it, to produce F(E), is equal to where R denotes
4T the resultant force of M, at the point E.
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 separate 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 my, 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 S,
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 my; 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.
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
7(72 – a ) the whole mass of the bar being denoted by m, its length by 2a, and A'C+ AC by 21. We conclude that a distribution of matter over the surface of the ellipsoid, having
47 1(22 – a) 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 gś 536, 537.
526. As a second example, let M consist of two equal particles, at points 1, I'. If we take the mass of each as unity, the potential at
1 Pis iet
and therefore IP'I'P
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 0 to 0, 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, 4.5, 4:3, 4:2, 4:1, 4, 3.9, 3:8, 307, 3.5, 3, 2:5, 2; the value of Il' being unity.
The corresponding equipotential surfaces are the surfaces traced by these curves, if the whole diagram is made to rotate round Il' as
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
1 surfaces, let R denote the resultant of forces equal to and
IP2 I'P2 in the lines PI and PI'. Then if matter be distributed over this
R surface, with density at P equal to its attraction on any internal
47' 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
R law of density, we have a shell of matter which exerts ($ 525)
on external points the same force as l'; and on internal points a force equal and opposite to that of I.
528. As an example of exceedingly great importance in the theory of electricity, let M consist of a positive mass, m, concentrated at a point I, and a negative mass, — m', at I'; and let s be a spherical surface cutting II', and Il' produced in points A, A,, such that IA: Al':: 1A,: I'A,::m:m'.
I Then, by a well-known geometrical proposition, we shall have IE: I'E::m : m'; and therefore
IE ΙΕ ́ Hence, by what we have just seen, one and the same distribution of matter over S will produce the same force as m' through all external space, and the same as m through all the space within S. And,
m' finding the resultant of the forces in El, and
in I'E, pro
I'E 2 duced, which, as these forces are inversely as IE to l'E, is ($ 222) equal to
maII 1 IEP.I' E
m IE3' we conclude that the density in the shell at E is
mo II 1
4 m IE: That the shell thus constituted does attract external points as if its mass were collected at I', and internal points as a certain mass collected at I, was proved geometrically in § 491 above. 529. If the spherical surface is given, and one of the points, 1, I',
CAP for instance I, the other is found by taking CI'= mass to be placed at it we have
I'A СА CI' m = m
ΑΙ СІ CA Hence, if we have any number of particles my, my, etc., at points I1, I2, etc., situated without S, we may find in the same way corresponding internal points l'1, I'z, etc., and masses m'1, m'z, etc.; and, by adding the expressions for the density at E given for each pair by the preceding formula, we get a spherical shell of matter which has the property of acting on all external space with the same force as -m'i, -m', etc., and on all internal points with a force equal and opposite to that of my, my, etc.
CI ; and for the