on particle (d) A right cone, of semivertical angle a, and length 1, Right cone attracts a particle at its vertex. Here we have at once for the at vertex. attraction, the expression 2πpl (1-cos α), 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. and discs. (e) For magnetic and electro-magnetic applications a very Positive useful case is that of two equal discs, each perpendicular to the negative line joining their centres, on any point in that line their masses (§ 461) 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 action is evidently In the particular case when c is diminished without limit, this becomes a2 force in crossing an attracting 478. Let P and P' be two points infinitely near one another variation of on two sides of a surface over which matter is distributed; and let p be the density of this distribution on the surface in the surface. 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 [§ 477, (b)] attract P and P with forces, each equal to 2πp and opposite to one another in the line PP'. Whence the proposition. It is one of much importance in the theory of electricity. Uniform hemisphere attracting particle at D edge. (a) As a further example of the direct analytical process, let P us find the components of the attraction exerted by a uniform hemisphere on a particle at its edge. Let A be the particle, AB a diameter of B the base, AC the tangent to the base at A; and AD perpendicular to AC, and AB. Let RQA be a section by a plane passing through AC; AQ any radius-vector of this section; Pa point in AQ. Let AP=r, CAQ=0, RAB=4. The volume of an element at P is rder sin Odp. dr r sin edo do dr. The resultant attraction on unit of matter at A has zero component along AC. Along AB the component is PЛff sin døde dr cos & sin 0, between proper limits. The limits of r are 0 and 2a sin 0 cos p, π those of are 0 and and those of 0 are 0 and π. Hence, 2' Alteration of latitude; by hemispherical hill or cavity. (b) Hence 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 πρα G-4 pa 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 Alteration a 2α R of latitude; by hemihill or south of it, differ by (2 + 2); ; instead of by as they would spherical 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 pa by where p is the density of the superficial strata. (c) For mutual attraction between two segments of a homogeneous solid sphere, investigated indirectly on a hydrostatic principle, see § 753 below. cavity, 479. As a curious additional example of the class of ques- by crevasse. tions considered in § 478 (a) (b), a deep crevasse, extending east and west, increases the latitude of places at its southern edge ρα 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 >1, 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. The reader may exercise himself by drawing lines of equal latitude in the neighbourhood of the crevasse in this case and by drawing meridians for the corresponding case of a crevasse running north and south. of a sphere 480. It is interesting, and will be useful later, to consider Attraction as a particular case, the attraction of a sphere whose mass is composed of composed of concentric layers, each of uniform density. concentric shells of uniform Let R be the radius, r that of any layer, p = F (r) its density, density. Then, if σ be the mean density, Attraction of a sphere composed of concentric shells of uniform density. If it is to be the same for all points inside the sphere If the density of the upper crust be r, the attraction at a depth h, small compared with the radius, is where σι is the mean density of nucleus when a shell of thickness h is removed from the sphere. Also, evidently, The attraction is therefore unaltered at a depth h if Attraction of a uniform circular arc, 481. Some other simple cases may be added here, as their results will be of use to us subsequently. C (a) The attraction of a circular arc, AB, of uniform density, B T 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 mass of element at P CD2 cos. PÓD Now suppose the density of the chord AB to be the same as that of the arc. Then for (mass of element at P x cos PCD) we may put mass of projection of element on AB at Q; since, if PT be the tangent at P, PTQ = PCD. Hence attraction along CD Sum of projected elements = PAB if p be the density of the given arc, Attraction of a uniform circular arc, 2p sin ACD = CD 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. line. (6) Again a limited straight line of uniform density attracts straight any external point in the same direction and with the same force as the corre sponding arc of a circle of the same density, which has the point for centre, and touches the straight line. For if CpP be A B C drawn cutting the circle in p and the line in P; Element at CP p: element at P :: Cp: CP that is, as Cp CP. 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 |