Attraction of a spheroid of revolution. a constant. When the attracted point lies on the surface of ellipsoid, the only requisite change is to put a=0. If we put cb, the ellipsoid becomes a spheroid of rever tion and for its attraction parallel to the axis we have : This integral is, of course, easily expressed in finite terns But, as we shall see presently, it is sufficient to find its value fr a point on the surface; for which we have 0 d(42) To work this out in real finite terms for an oblate spheroid, let 6 For one of the components perpendicular to the axis we have theorem 523. From what we have already given of the analysis of this question, it is easy to deduce the following splend: i theorem, due to Maclaurin: - The attractions exerted by two homogeneous and confocal e'l p theorem. soids on the same point external to each, or external to one and Maclaurin's on the surface of the other, are in the same direction and proportional to their masses. If we put h+2 for 2 in the first integral and in the equation. 524. In a similar way we may at once prove Ivory's Ivory's theorem 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. The component of M on §, 7. §. is already given [§ 522 (5)]. + + =1. 2 or 2 a2+h+p2 ̄ ̄b2+h+ø2 Now the integrals are evidently equal-and the whole expres sions are as M to M, do; i.e., as b。c。 to b‚ê‚. k theorem. Ivory's theorem. Law of attraction 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 (§ 519). 525. An ingenious application of Ivory's theorem, by when a uni- Duhamel, must not be omitted here. Concentric spheres are a particular case of confocal ellipsoids, and therefore the ataction on an traction of any sphere on a point on the surface of an internal form spheri cal shell exerts no internal point. Centre of gravity. 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. 526. (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 Centrobaric a centrobaric body. bodies, proved possible by Green. 527. 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. Properties of 528. If a body (B) is centrobaric relatively to any one centrobaric bodies. attracting mass (4), 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.1 Let O be any point so distant from B that a spherical surface described from it as centre, and not containing any part of B, is large enough entirely to contain A. Let A be placed within any such spherical surface and made to rotate about any axis, OK, through O. It will always attract B in a line through G, the centre of gravity of B. Hence if every particle of its mass 1 Thomson, Proc. R.S.E., Feb. 1864. centrobaric be uniformly distributed over the circumference of the circle Properties of that it describes in this rotation, the mass, thus obtained, will bodies. also attract B in a line through G. And this will be the case however this mass is rotated round 0; since before obtaining it we might have rotated A and OK in any way round O, holding them fixed relatively to one another. We have therefore found a body, A', symmetrical about an axis, OK, relatively to which B is necessarily centrobaric. Now, O being kept fixed, let OK, carrying A' with it, be put successively into an infinite number, n, of positions uniformly distributed round 0; that is to say, so that there are equal numbers of positions of OK in all equal solid angles round : and let n part of the mass of A' be left in each of the positions into which it was thus necessarily carried. B will experience from A all this distribution of matter, still a resultant force through G. But this distribution, being symmetrical all round 0, consists of uniform concentric shells, and (§ 471) the mass of each of these shells might be collected at 0 without changing its attraction on any particle of B, and therefore without changing its resultant attraction on B. Hence B is centrobaric relatively to a mass collected at 0; this being any point whatever not nearer than within a certain limiting distance from B (according to the condition stated above). That is to say, any point. placed beyond this distance is attracted by B in a line through. G; and hence, beyond this distance, the equipotential surfaces of B are spherical with G for common centre. B therefore attracts points beyond this distance as if its mass were collected at G and it follows (§ 497) that it does so also through the whole space external to itself. Hence it attracts any group of points, or any mass whatever, external to it, as if its own mass were collected at G. 529. Hence §§ 497, 492 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, each in space external to all: in other words, the outer boundary of every centrobaric body is a single closed surface. centrobaric bodies. Properties of 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. Centrobaric shell. 530. 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 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 proved above (§ 499). 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:- 531. 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 §§ 501, 506 above, it appears also that a centrobaric shell may be made of either half of the lemniscate in the diagram of $508, 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 § 506, |