48. Having given the positions of the sun and moon supposed to be in the equator, find the interval between high water at any place and the passage of the moon over the meridian. 49. Investigate the change in the qinclination of the ecliptic produced by the sun's action on the earth. 50. Find the proportion of the axis of a homogeneous revolving fluid spheroid ; shew that there are two forms of equilibrium, and find the eccentricity in each case when the centrifugal force is small. 51. If several angular velocities be impressed on a body at the same time, the resulting axis of rotation, and the angular velocity about it, will be found by finding the direction and magnitude of the resultant of forces in the directions of the several axes of rotation, and proportional to the angular velocities. 52. Find the attraction of a spherical shell on a particle 1 without it, the law of attraction being (dist.) 53. If the earth be a solid of revolution, such that r is the radius of curvature of its meridian at the equator, and r t u sin.1 the radius of curvature of its meridian in any latitude 1, prove that the polar and equatorial axes are respectively 1 8 rt 15 и M. 1832 and pt 5 Mo 54. Determine the height to which the disturbing force of the sun would raise the waters of the earth above the equicapacious sphere, supposing the earth to be at rest and to be covered with fluid. 55. Find the attraction of a spherical shell on any point without it, the force of attraction of each particle varying as 1 and hence determine the attraction of a homogeneous (dist.)? sphere. 56. If CA be the radius of a sphere, the centre of which is C, and Q, P two points in the same diameter, Q within and 1833 P without the sphere, so situated that CQ, CP = CA?, and if the attractive force of each particle of the sphere oc (distance)-", prove that the attraction on Q: attraction on P :: VQC.PC: VPC.QCM. 57. Prove that the attraction of an oblate spheroid on a particle at any point in its surface in a direction perpendicular to the axis, is proportional to the distance of the particle from the axis, and in a direction perpendicular to the equator to its distance from the equator. 58. The tide produced at the equator by the joint action of the sun and moon, being }{h sin’8 cos 2 (0 - 1) + h sin’cos 2 (0 – %')}, where 6, 6 are the respective hour angles of the sun and moon, determine the time and height of high water. On what suppositions is the above formular obtained ? 59. The earth being supposed a homogeneous oblate spheroid, find the angular velocity generated in an indefinitely small time round an equatorial axis by the action of the sun. 60. Prove that the attraction of a homogeneous sphere on a particle without it, is the same as if all the mass were collected in its centre; force varying inversely as the square of the distance. Is this result true for any other laws of force? 61. What is the practical method of observing the number 1834 of vibrations made by a pendulum in 24 hours ? Shew how the mass of the earth may be determined by a comparison of the rates of oscillation of a pendulum at the surface, and at a point below the surface. 62. Shew that velocity of the earth's rotation is unaltered by the action of the sun and moon, first proving that the points of intersection of the axis of instantaneous rotation with a sphere described about the kearth's centre lie nearly in a small circle. 63. By what experiments is it shewn that terrestrial bodies are attracted by the earth and by each other proportionably to their quantities of matter? 1835 64. Compare the retardation of the tide at syzygy and quadrature. What is meant by the establishment of a port? 65. Having given the attractions of an oblate spheroid of small eccentricity upon particles at its pole and equator, express the ellipticity of a spheroid of equilibrium revolving slowly, in terms of the ratio of the centrifugal force at its equator to gravity. How may we compare the ellipticities of two planets which have satellites, supposing them homogeneous ? 66. The pole of the earth, affected only by the sun's attraction, traces out a curve on the celestial sphere, whose equation is tan i P . 1 s 67. A particle placed on the inner surface of a spheroidal shell, bounded by similar concentric spheroidal surfaces, will remain at rest, supposing the force of attraction of each particle to vary inversely as the square of the distance. n 68. Compare the retardation of the spring and neap tides; and find the time of high water when the moon is syzygy. 69. Find the attraction of an oblate spheroid on a particle situated at its pole. 70. Find the law of attraction, that a sphere may attract a particle without it in the same manner as if the whole mass were collected in the centre. 71. Assuming the attraction of a homogeneous sphervid on any point to be known, enunciate all the propositions necessary for finding the attraction of a heterogeneous spheroid on a point within it; and supposing the latter given, and the density of equally dense spheroidal surfaces a known function of the polar distance, give the train of reasoning from which the ellipticity of the earth may be determined. 72. If a uniform force acting upon a body tend to give it a motion of rotation round an axis which is always perpen dicular to the axis round which it is at each instant revolving, and the axes be always in the same plane, the angular velocity will be unaltered. 73. State the theory of universal gravitation, and point out generally the evidence on which it has been received. How is it shewn that bodies are attracted towards the earth by forces tending to each part of the mass ? 74. A sphere, whose surface is perfectly smooth, by its 1836 attraction keeps attached to it the point of a needle, the other end of which rests upon a perfectly smooth horizontal plane, situated below the sphere at a distance less than the length of the needle. Suppose the needle originally not placed in a position of equilibrium, determine the nature of its motion. 75. Determine the attraction of a right prism of square base on a particle in its axis, the force to each particle varying inversely as the square of the distance. 76. Find the attraction of a sphere upon a point, and thence that of one sphere upon another; the force varying inversely as the square of the distance. 77. Supposing the earth to have been originally a homogeneous fluid mass revolving uniformly about its axis, shew that there will be equilibrium when the force at the pole is to that at the equator as the radius of the equator is to that of the pole. 78. Investigate the motion of the pole of the earth produced by the moon in one sidereal revolution. face e 79. The equation for determining the ellipticities of the spheroids of equal density in the earth is d(ce) dc 1 dc - Sc=c} pd.ec3 505 2T2 р its solution may be made to d2v 6v depend on that of the equation de2 c2 + q?v = 0. pc? {cze 0; с =c{(1 {(1) c)} Having given the complete solution of this 3 3 V = C sin (qc + C') + cos (gc + q?c? qc determine the earth's ellipticity at the surface. 80. When electricity is latent in any body, the quantity of positive electricity contained in it is equal to that of negative. 81. Shew that the action of the moon produces contemporaneous tides on opposite sides of the earth. At a given place find the elevation of the tide caused by the joint action of the sun and moon, both supposed to be in the equator. 1 |