and equally inclined to the axis of the cone, exert equal forces on a particle at the vertex. For the area of any inclined section, whatever be its orientation, is greater than that of the corresponding transverse section in the ratio of unity to the cosine of the angle of inclination. Hence if a plane touch a sphere at a point B, and if the plane and sphere have equal surface density at corresponding points P and p in a line drawn through A, the point diametrically opposite to B, corresponding elements at P and p exert equal attraction on a particle at A. Thus the attraction on A, of any part of the plane, is the same as that of the corresponding part of the sphere, cut out P P B A by a cone of infinitely small angle whose vertex is A. Hence if we resolve along the line AB the attraction of pq on A, the component is equal to the attraction along Ap of the transverse section pr, i. e. pw, where w is the angle subtended at A by the element pq, and p the surface density. ρ Thus any portion whatever of the sphere attracts A along AB with a force proportional to its spherical opening as seen from A; and the same is, by what was proved above, true of a flat plate. Hence as a disc of radius a subtends at a point distant h from it, in the direction of the axis of the disc, a spherical angle which for an infinite disc becomes, whatever the distance h, 2 πρ. From the preceding formula many useful results may easily be deduced: thus, (6) A uniform cylinder of length 1, and diameter a, attracts a point in its axis at a distance x from the nearest end with a force 2πρ {l— √(x+1)2 + a2 + √x2 + a2}. When the cylinder is of infinite length (in one direction) the attraction is therefore and, when the attracted particle is in contact with the centre of the end of the infinite cylinder, this is 2πρα. (c) A right cone, of semivertical angle a, and length 7, attracts a particle at its vertex. Here we have at once for the attraction, the expression which is simply proportional to the length of the axis. It is of course easy, when required, to find the necessarily less simple expression for the attraction on any point of the axis. (d) For magnetic and electro-magnetic applications a very useful case is that of two equal uniform discs, each perpendicular to the line joining their centres, on any point in that line—their masses (§ 478) being of opposite sign—that is, one repelling and the other attracting. Let a be the radius, p the mass of a superficial unit, of either, c their distance, x the distance of the attracted point from the nearest disc. The whole force is evidently In the particular case when c is diminished without limit, this becomes a2 2 προ 495. Let P and P' be two points infinitely near one another on two sides of a surface over which matter is distributed; and let ρ be the density of this distribution on the surface in the neighbourhood of these points. Then whatever be the resultant attraction, R, at P, due to all the attracting matter, whether lodging on this surface, or elsewhere, the resultant force, R', on P' is the resultant of a force equal and parallel to R, and a force equal to 4πp, in the direction from P' perpendicularly towards the surface. For, suppose PP' to be perpendicular to the surface, which will not limit the generality of the proposition, and consider a circular disc, of the surface, having its centre in PP', and radius infinitely small in comparison with the radii of curvature of the surface but infinitely great in comparison with PP'. This disc will [§ 494] attract P and P' with forces, each equal to 2πр and opposite to one another in the line PP'. Whence the proposition. It is one of much importance in the theory of electricity. 496. It may be shown that at the southern base of a hemispherical hill of radius a and density p, the true latitude (as measured by the aid of the plumb-line, or by reflection of starlight in a trough of mercury) is diminished by the attraction of the mountain by the angle έρπα where G is the attraction of the earth, estimated in the same units. Hence, if R be the radius and σ the mean density of the earth, the angle is Hence the latitudes of stations at the base of the hill, north and a south of it, differ by (2 + 2); instead of by R do if the hill were removed. In the same way the latitude of a place at the southern edge of a hemispherical cavity is increased on account of the cavity by where p is the density of the superficial strata. ρα σα ρα σR 497. As a curious additional example of the class of questions we have just considered, a deep crevasse, extending east and west, increases the latitude of places at its southern edge by (approximately) the angle where ρ is the density of the crust of the earth, and a is the width of the crevasse. Thus the north edge of the crevasse will have a lower latitude than the south edge if 2 which might be the case, as there are rocks of density × 55 or 3.67 times that of water. At a considerable depth in the crevasse, this change of latitudes is nearly doubled, and then the southern side has the greater latitude if the density of the crust be not less than 1.83 times that of water. σ > 1, 498. It is interesting, and will be useful later, to consider as a particular case, the attraction of a sphere whose mass is composed of concentric layers, each of uniform density. Let σ be, as above, the mean density of the whole globe, and τ the density of the upper crust. The attraction at a depth h, small compared with the radius, is where 1 denotes the mean density of the nucleus remaining when a shell of thickness h is removed from the sphere. Also, evidently, Tσ, (R-h)+4πт (R―h)2h=πσR3, or whence G1(R − h)2 + 4πт(R − h)2h=GR2, G1=G(1+2) — 4.Th. The attraction is therefore unaltered at a depth h if 499. Some other simple cases may be added here, as their results will be of use to us subsequently. (a) The attraction of a circular arc, AB, of uniform density, on a particle at the centre, C, of the circle, lies evidently in the line CD bisecting the arc. Also the resolved part parallel to CD of the attraction of an element at P is It is therefore the same as the attraction of a mass equal to the chord, with the arc's density, concentrated at the point D. (6) Again, a limited straight line of uniform density attracts any external point in the same direction and with the same force as the corresponding arc of a circle of the same density, which has the point for centre, and touches the straight line. For if CpP be drawn cutting the circle in p and the line in P; ele that is, as Cp: CP2. Hence the attractions of these elements on C are equal and in the same line. Thus the arc ab attracts C as the line AB does; and, by the last proposition, the attraction of AB bisects the angle ACB, and is equal to 2p = PAB (AC+CB) CK2(AC+CB)(AC÷CB2—AB2) For, evidently, bK: Ka:: BK: KA:: BC: CA:: bC: Ca, CF. (1) i. e., ab is divided, externally in C, and internally in K, in the same ratio. Hence, by geometry, KC.CF aC.Cb=1{AC+CB2-AB2}, which gives the transformation in (1). (d) CF is obviously the tangent at C to a hyperbola, passing through that point, and having A and B as foci. Hence, if in any plane through AB any hyperbola be described, with foci A and B, it will be a line of force as regards the attraction of the line AB; that is, as will be more fully explained later, a curve which at every point indicates the direction of attraction. (e) Similarly, if a prolate spheroid be described with foci A and B, and passing through C, CF will evidently be the normal at C; thus the force on a particle at C will be perpendicular to the spheroid; and the particle would evidently rest in equilibrium on the surface, even if it were smooth. This is an instance of (what we shall presently develop at some length) a surface of equilibrium, a level surface, or an equipotential surface. (f) We may further prove, by a simple application of the preceding theorem, that the lines of force due to the attraction of two infinitely long rods in the line AB produced, one of which is attractive and the other repulsive, are the series of ellipses described from the extremities, A and B, as foci, while the surfaces of equilibrium are generated by the revolution of the confocal hyperbolas. |