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530. An infinite number of such particles may be given, constituting a continuous mass M; when of course the corresponding internal particles will constitute a continuous mass, – M', of the opposite kind of matter; and the same conclusion will hold. If S is the surface of a solid or hollow metal ball connected with the earth by a fine wire, and M an external influencing body, the shell of matter we have determined is precisely the distribution of electricity on S called out by the influence of M: and the mass – M', determined as
above, is called the Electric Image of M in the ball, since the electric | action through the whole space external to the ball would be
unchanged if the ball were removed and – M' properly placed in the space left vacant. We intend to return to this subject under Electricity.
531. Irrespectively of the special electric application, this method of images gives a remarkable kind of transformation which is often useful. It suggests for mere geometry what has been called the transformation by reciprocal radius-vectors; that is to say, the substitution for any set of points, or for any diagram of lines or surfaces, another obtained by drawing radii to them from a certain fixed point or origin, and measuring off lengths inversely proportional to these radii along their directions. We see in a moment by elementary geometry that any line thus obtained cuts the radius-vector through any point of it at the same angle and in the same plane as the line from which it is derived. Hence any two lines or surfaces that cut one another give two transformed lines or surfaces cutting at the same angle : and infinitely small lengths, areas, and volumes transform into others whose magnitudes are altered respectively in the ratios of the first, second, and third powers of the distances of the latter from the origin, to the same powers of the distances of the former from the same. Hence the lengths, areas, and volumes in the transformed diagram, corresponding to a set of given equal infinitely small lengths, areas, and volumes, however situated, at different distances from the origin, are inversely as the squares, the fourth powers and the sixth powers of these distances. Further, it is easily proved that a straight line and a plane transform into a circle and a spherical surface, each passing through the origin ; and that, generally, circles and spheres transform into circles and spheres.
532. In the theory of attraction, the transformation of masses, densities, and potentials has also to be considered. Thus, according to the foundation of the method ($ 530), equal masses, of infinitely small dimensions at different distances from the origin, transform into masses inversely as these distances, or directly as the transformed distances : and, therefore, equal densities of lines, of surfaces, and of solids, given at any stated distances from the origin, transform into densities directly as the first, the third, and the fifth powers of those distances; or inversely as the same powers of the distances, from the origin, of the corresponding points in the transformed system. The usefulness of this transformation in the theory of electricity,
and of attraction in general, depends entirely on the following theorem:
Let o denote the potential at P due to the given distribution, and the potential at P due to the transformed distribution: then shall
D'==$=$. Let a mass m collected at I be any part of the given distribution, and let m'at I'be the corresponding part in
P the transformed distribution.
PL We have
ai=O1'.Ol=OP.OP, and therefore
I OI: OP :: OPP: 01'; which shows that the triangles IPO, P'I'O are similar, so that
IP:P'I':: VOI.OP: JOP.OI':: 01.0P:a. We have besides
m:m' :: 01:a, and therefore
:: OP: a. IPI'P
Hence each term of $ bears to the corresponding term of $' the same ratio; and therefore the sum, 0, must be to the sum, $, in that ratio, as was to be proved.
533. As an example, let the given distribution be confined to a spherical surface, and let O be its centre and a its own radius. The transformed distribution is the same. But the space within it becomes transformed into the space without it. Hence if o be the potential due to any spherical shell at a point P, within it, the potential due
a? to the same shell at the point P' in OP produced till OP'= is
OP equal to Opo (which is an elementary proposition in the spherical harmonic treatment of potentials, as we shall see presently). Thus, for instance, let the distribution be uniform. Then, as we know there is no force on an interior point, must be constant; and therefore the potential at P', any external point, is inversely proportional to its distance from the centre.
Or let the given distribution be a uniform shell, S, and let O be any eccentric or any external point. The transformed distribution becomes ($$ 531, 532) a spherical shell, S, with density varying inversely as the cube of the distance from 0. If O is within S, it is also enclosed by S', and the whole space within S transforms into
the whole space without S'. Hence ($ 532) the potential of Sat any point without it is inversely as the distance from 0, and is therefore that of a certain quantity of matter collected at 0. Or if O is external to S, and consequently also external to S', the space within S transforms into the space within S'. Hence the potential of S' at any point within it is the same as that of a certain quantity of matter collected at 0, which is now a point external to it. Thus, without taking advantage of the general theorems ($9 517, 524), we fall back on the same results as we inferred from them in § 528, and as we proved synthetically earlier (89 488, 491, 492). It may be remarked that those synthetical demonstrations consist merely of transformations of Newton's demonstration, that attractions balance on a point within a uniform shell. Thus the first of them ($ 488) is the image of Newton's in a concentric spherical surface; and the second is its image in a spherical surface having its centre external to the shell, or internal but eccentric, according as the first or the second diagram is used.
534. We shall give just one other application of the theorem of $ 532 at present, but much use of it will be made later in the theory of Electricity.
Let the given distribution of matter be a uniform solid sphere, B, and let O be external to it. The transformed system will be a solid sphere, B', with density varying inversely as the fifth power of the distance from O, a point external to it. The potential of S is the same throughout external space as that due to its mass, m, collected at its centre, C. Hence the potential of S' through space external to it is the same as that of the corresponding quantity of matter collected at C', the transformed position of C. This quantity is of course equal to the mass of B. And it is easily proved that C'is the position of the image of 0 in the spherical surface of B'. We conclude that a solid sphere with density varying inversely as the fifth power of the distance from an external point, 0, attracts any external point as if its mass were condensed at the image of 0 in its external surface. It is easy to verify this for points of the axis by direct integration, and thence the general conclusion follows according to $ 508.
535. The determination of the attraction of an ellipsoid, or of an ellipsoidal shell, is a problem of great interest, and its results will be of great use to us afterwards, especially in Magnetism. We have left it till now, in order that we may be prepared to apply the properties of the potential, as they afford an extremely elegant method of treatment. A few definitions and lemmas are necessary.
Corresponding points on two confocal ellipsoids are such as coincide when either ellipsoid by a pure strain is deformed so as to coincide with the other.
And it is easily shown, that if any two points, P, Q, be assumed on one shell, and their corresponding points, p, q, on the other, we have Pq=lp.
The species of shell which it is most convenient to employ in the subdivision of a homogeneous ellipsoid is bounded by similar, similarly situated, and concentric ellipsoidal surfaces; and it is evident from the properties of pure strain ($ 141) that such a shell may be produced from a spherical shell of uniform thickness by uniform extensions and compressions in three rectangular directions. Unless the contrary be specified, the word 'shell' in connection with this subject will always signify an infinitely thin shell of the kind now described.
536. Since, by § 479, a homogeneous spherical shell exerts no attraction on an internal point, a homogeneous shell (which need not be infinitely thin) bounded by similar, and similarly situated, and concentric ellipsoids, exerts no attraction on an internal point.
For suppose the spherical shell of $ 479, by simple extensions and compressions in three rectangular directions, 'to be transformed into an ellipsoidal shell. In this distorted form the masses of all parts are reduced or increased in the proportion of the mass of the ellipsoid to that of the sphere. Also the ratio of the lines HP, PK is unaltered, § 139. Hence the elements IH, KL still attract P equally, and the proposition follows as in $ 479.
Hence inside the shell the potential is constant.
537. Two confocal shells ($ 535) being given, the potential of the first at any point, P, of the surface of the second, is to that of the second at the corresponding point, p, on the surface of the first, as the mass of the first is to the mass of the second. This beautiful proposition is due to Chasles.
To any element of the mass of the outer shell at l corresponds an element of mass of the inner at q, and these bear the same ratio to the whole masses of their respective shells, that the corresponding element of the spherical shell from which either may be derived bears to its whole mass. Whence, since Pq=Qp, the proposition is true for the corresponding elements at 7 and q, and therefore for the entire shells.
Also, as the potential of a shell on an internal point is constant, and as one of two confocal ellipsoids is wholly within the other : it follows that the external equipotential surfaces for any such shell are confocal ellipsoids, and therefore that the attraction of the shell on an external point is normal to a confocal ellipsoid passing through the point.
538. Now it has been shown (§ 495) that the attraction of a shell on an external point near its surface exceeds that on an internal point infinitely near it by 4mp where p is the surface-density of the shell at that point. Hence, as ($ 536) there is no attraction on an internal point, the attraction of a shell on a point at its exterior surface is 4tp: or 4 Tpt if p be now put for the volume-density, and t for the (infinitely small) thickness of the shell, § 495. From this it is easy to obtain by integration the determination of the whole attraction of a homogeneous ellipsoid on an external particle.
539. The following splendid theorem is due to Maclaurin :
The attractions exerted by two homogeneous and confocal ellipsoids on the same point external to each, or external to one and on the surface of the other, are in the same direction and proportional to their
540. Ivory's theorem is as follows :
Let corresponding points P, p, be taken on the surfaces of two homogeneous confocal ellipsoids, E, e. The x component of the attraction of E on p, is to that of e on P as the area of the section of E by the plane of yz is to that of the coplanar section of e.
Poisson showed that this theorem is true for any law of force whatever. This is easily proved by employing in the general expressions for the components of the attraction of any body, after one integration, the properties of corresponding points upon confocal ellipsoids ($ 535).
541. An ingenious application of Ivory's theorem, by Duhamel, must not be omitted here. Concentric spheres are a particular case of confocal ellipsoids, and therefore the attraction of any sphere on a point on the surface of an internal concentric sphere, is to that of the latter upon a point in the surface of the former as the squares
of the radii of the spheres. Now if the law of attraction be such that a homogeneous spherical shell of uniform thickness exerts no attraction on an internal point, the action of the larger sphere on the internal point is reducible to that of the smaller. Hence the law is that of the inverse square of the distance, as is easily seen by making the smaller sphere less and less till it becomes a mere particle. This theorem is due originally to Cavendish.
542. (Definition.) If the action of terrestrial or other gravity on a rigid body is reducible to a single force in a line passing always through one point fixed relatively to the body, whatever be its position relatively to the earth or other attracting mass, that point is called its centre of gravity, and the body is called a centrobaric body.
543. One of the most startling results of Green's wonderful theory of the potential is its establishment of the existence of centrobaric bodies; and the discovery of their properties is not the least curious and interesting among its very various applications.
544. If a body (B) is centrobaric relatively to any one attracting mass (A), it is centrobaric relatively to every other : and it attracts all matter external to itself as if its own mass were collected in its centre of gravity.
545. Hence &$ 510, 515 show that,
(a) The centre of gravity of a centrobaric body necessarily lies in its interior ; or in other words, can only be reached from external space by a path cutting through some of its mass. And
(6) No centrobaric body can consist of parts isolated from one another,
1 Thomson. Proc. R.S.E., Feb. 1864.