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 connexion 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 Q corresponds an element of mass of the inner at 9, 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 Q and 9, 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 4πp 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 4p: or 4mpt if p be now put for the volume-density, and 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. 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 masses. 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 (b) No centrobaric body can consist of parts isolated from one another, 1 Thomson. Proc. R. S. E., Feb. 1864. each in space external to all: in other words, the outer boundary of every centrobaric body is a single closed surface. Thus we see, by (a), that no symmetrical ring, or hollow cylinder with open ends, can have a centre of gravity; for its centre of gravity, if it had one, would be in its axis, and therefore external to its mass. 546. If any mass whatever, M, and any single surface, S, completely enclosing it be given, a distribution of any given amount, M', of matter on this surface may be found which shall make the whole centrobaric with its centre of gravity in any given position (G) within that surface. The condition here to be fulfilled is to distribute M' over S, so as by it to produce the potential at any point, E, of S; V denoting the potential of M at this point. The possibility and singleness of the solution of this problem were stated above (§ 517). It is to be remarked, however, that if M' be not given in sufficient amount, an extra quantity must be taken, but neutralized by an equal quantity of negative matter, to constitute the required distribution on S. The case in which there is no given body M to begin with is important; and yields the following: 547. A given quantity of matter may be distributed in one way, but in only one way, over any given closed surface, so as to constitute a centrobaric body with its centre of gravity at any given point within it. Thus we have already seen that the condition is fulfilled by making the density inversely as the distance from the given point, if the surface be spherical. From what was proved in $$ 519, 524 above, it appears also that a centrobaric shell may be made of either half of the lemniscate in the diagram of § 526, or of any of the ovals within it, by distributing matter with density proportional to the resultant force of m at I and m' at I'; and that the one of these points which is within it is its centre of gravity. And generally, by drawing the equipotential surfaces relatively to a mass m collected at a point I, and any other distribution of matter whatever not surrounding this point; and by taking one of these surfaces which encloses I but no other part of the mass, we learn, by Green's general theorem, and the special proposition of § 524, how to distribute matter over it so as to make it a centrobaric shell with I for centre of gravity. 548. Under hydrokinetics the same problem will be solved for a cube, or a rectangular parallelepiped in general, in terms of converging series; and under electricity (in a subsequent volume) it will be solved in finite algebraic terms for the surface of a lens bounded by two spherical surfaces cutting one another at any sub-multiple of two right angles, and for either part obtained by dividing this surface in two by a third spherical surface cutting each of its sides at right angles. 549. Matter may be distributed in an infinite number of ways throughout a given closed space, to constitute a centrobaric body with its centre of gravity at any given point within it. For by an infinite number of surfaces, each enclosing the given point, the whole space between this point and the given closed surface may be divided into infinitely thin shells; and matter may be distributed on each of these so as to make it centrobaric with its centre of gravity at the given point. Both the forms of these shells and the quantities of matter distributed on them, may be arbitrarily varied in an infinite variety of ways. Thus, for example, if the given closed surface be the pointed oval constituted by either half of the lemniscate of the diagram of § 526, and if the given point be the point I within it, a centrobaric solid may be built up of the interior ovals with matter distributed over them to make them centrobaric shells as above ($ 547). From what was proved in § 534, we see that a solid sphere with its density varying inversely as the fifth power of the distance from an external point, is centrobaric, and that its centre of gravity is the image (§ 530) of this point relatively to its surface. 550. The centre of gravity of a centrobaric body composed of true gravitating matter is its centre of inertia. For a centrobaric body, if attracted only by another infinitely distant body, or by matter so distributed round itself as to produce (§ 517) uniform force in parallel lines throughout the space occupied by it, experiences (§ 544) a resultant force always through its centre of gravity. But in this case this force is the resultant of parallel forces on all the particles of the body, which (see Properties of Matter, below) are rigorously proportional to their masses: and it is proved that the resultant of such a system of parallel forces passes through the point defined in $ 195, as the centre of inertia. 551. The moments of inertia of a centrobaric body are equal round all axes through its centre of inertia. In other words (§ 239), all these axes are principal axes, and the body is kinetically symmetrical round its centre of inertia. CHAPTER VII. STATICS OF SOLIDS AND FLUIDS. B 552. FORCES Whose lines meet. Let ABC be a rigid body acted on by two forces, P and Q, applied to it at different points, D and E respectively, in lines in the same plane. Since the lines are not parallel, they will meet if produced; let them be produced and meet in O. Transmit the forces to act on that point; and the result is that we have simply the case of two forces acting on a material point, which has been already considered. 553. The preceding solution is applicable to every case of nonparallel forces in a plane, however far removed the point may be in which their lines of action meet, and the resultant will of course be found by the parallelogram of forces. The limiting case of parallel forces, or forces whose lines of action, however far produced, do not meet, was considered above, and the position and magnitude of the resultant were investigated. The following is an independent demonstration of the conclusion arrived at. 554. Parallel forces in a plane. The resultant of two parallel forces is equal to their sum, and is in the parallel line which divides any line drawn across their lines of action into parts inversely as their magnitudes. 1o. Let P and Q be two parallel forces acting on a rigid body in similar directions in lines AB and CD. Draw any line AC across their lines. In it introduce any SL pair of balancing forces, S in AG K-S M N |