dist.' 37. Explain fully the motion of the apsides, and the variation of the eccentricity of P's orbit, Cors. 8 and 9, Prop. 66. 38. If a body describing a spiral in a medium whose den 1 sity cut the radius vector at B in the same angle as at A, and with a velocity which is to that at A ::SA :VSB, then will the distances at which the body cuts the radius vector, and the times of successive revolutions, be in geometric progression. (Newton, Book II. Cor. 7, Prop. 15.) 39. Determine the angular distance of a body from the vertex of an ellipse whose eccentricity = }, at which the velocity : greatest velocity :: 1:73. 40. Shew that the velocity acquired by a body in falling from infinity to the earth's centre, is to the velocity of a secondary at the earth's surface :: 73:1. 41. Find where an inferior planet will appear stationary, supposing the force of gravity to vary inversely as the cube of the distance, and the orbits of the earth and planet to be circular. 42. In Cotes's first spiral, it is required to shew that the successive distances at which the curve cuts the apsidal line may be represented by a series of the form, c2 C3 c2 + 16A + 1 c + T' 43. To determine the mean motion of the moon's nodes. (Newton, Book III. Prop. 32.) bAm + cAn 44. Force varies as find the angle between the A3 apsides. 45. Explain the nature of centrifugal force, and shew that ha in all curves it where h = twice the area described in . one second of time. 46. Let a be the distance from the centre of force, from which a body must fall externally to acquire the velocity in a 1 circle whose radius = r when the force a dist. ; and let b be с сп can +1 1824 the distance to which it must fall internally to acquire the same velocity; then will a, r, 6 be in geometric progression. 47. If at the distance a from the centre of force, a body be projected at an angle of 45° with the distance, with a velocity which is to that in a circle at the same distance as v 2 to V3; 2a2n n and the force = + dist.5 ; required the curve described. dist.3 48. A body acted on by gravity moves on the surface of a cone whose axis is vertical; find the law of force tending to the vertex, by which the projection of its path on a horizontal plane passing through the vertex may be described ; and hence deduce the angle between the apsides when the orbit is nearly circular. 49. If a body describe the spiral of Archimedes, the force being in the pole, and its motion beginning from that point; then will the times of the successive revolutions be as the differences of the cubes of the natural numbers; and the excess of the time of the n + 1th revolution above that of the nth = nx excess of the 2nd above the 1st. the 50. If S be the sun, and A, B two planets that appear stationary to one another ; then tan SBA : tan SAB :: periodic time of A: periodic time of B. 51. If the periodic times of a body revolving in a circle round the centres of force S and R be the same, compare force tending to S with that tending to R. 52. If P' be a point so taken in the radius vector SP of a parabola, that SP = SY the perpendicular on the tangent, then will the locus of the point P' be the elliptic spiral; prove this, and compare the times of two bodies describing AP and AP, the absolute forces being the same in both cases, 53. If a body describe an oval round a centre of force, the distance at which the angular velocity is equal to the mean A angular velocity is where A is the area of the figure, and a = 3,14159. 54. An imperfectly elastic body revolving in an ellipse whose eccentricity is }, is reflected at the mean distance by a plane coincident with the distance, so as to move after impact in the direction of the axis minor; find the degree of elasticity, and compare the periodic times in the two ellipses (F« bа). 55. Find the horary motion of the moon's nodes in a circular orbit. 1825 56. Find the space through which a body must fall externally that it may acquire the velocity with which it moves in an ellipse about the centre. 57. If the mass of a planet is four times that of the earth, and the distance of its satellite 16 times that of the moon from the earth, in how many months will the satellite revolve ? 58. When the centripetal force varies inversely as the nth power of the distance, n being greater than 3; find the equation to the spiral, which a body, projected with a velocity equal to the velocity acquired by falling from an infinite distance, describes; and determine the number of revolutions which it makes, before it falls into the centre. 59. Compare the mean addititious force with the force by which P is retained in its orbit round T. (Newton, Prop. 66, Cor. 17.) 60. State the phenomena, from which it appears that the force by which the moon is retained in its orbit, tends to the earth, and that this force varies inversely as the square of the distance. 61. In the 10th Lemma of the 1st Section of Newton, where the abscissa AD represents the time, the ordinate DB the velocity, and the area ABD the space described ; if a straight line be drawn touching the curve AB in B the extremity of the ordinate, the tangent of the angle which this line makes with the axis will represent the force. Required proof. 62. Investigate the formula of lunar nutation in right ascension; find the longitude of the moon's node when it is a maximum, and thence its maximum value; and by means of these values express the nutation in right ascension in a more simple formula. 63. Determine generally the resistance of a medium, so that a body acted upon by a centripetal force, whose law is known, may move in a given curve; and thence find the resistance when the force is uniform and acts in parallel lines, so that the curve may be a circular arc. 64. Find the law of the force tending to the pole of the logarithmic spiral. 65. When the force varies inversely as the nth power of the distance, compare the velocity acquired by falling from an infinite distance with the velocity of a body revolving in a circle, 66. Find the point in a given hyperbola where the velocity of a body acted on by a force tending to its focus is twice as great as the velocity in a parabola at the same distance. 67. If the earth's motion about its axis were to cease, how much would a clock keeping true time in a given latitude gain in 24 hours ? 68. There is a fixed centre of force which varies inversely as the square of the distance; and about this as a focus an ellipse is described, the axes of which are to one another in the proportion of v 2:1. A perfectly elastic body falls from a distance equal to the axis-major in the direction of the radius vector passing through the extremity of the axis-minor and impinges on the ellipse. Required the motion after impact. 69. The earth being supposed a sphere revolving about its axis with a given angular velocity, find the curve, in a meridional plane, which is the locus of a body, the centrifugal force of which opposed to gravity is every where equal to the force of gravity acting upon it. 70. State the methods by which the masses of all the planets and of their satellites may be compared with that of the sun. 71. If a body be revolving in an ellipse about the focus, and the force be suddenly made to vary inversely as the cube of the distance, the actual force at the mean distance being unaltered, what will be the curve described ? 1826 72. Investigate the equation to the orbit in which the centripetal force is always n times as great as the centrifugal, and find the time of one revolution. 73. Determine that point in an ellipse described round a centre of force situated in the focus, where the linear velocity is n times as great as the paracentric. 74. If a body describe a logarithmic curve by a force acting perpendicularly to its axis; prove that the force at any point varies as the body's distance from the axis, and the velocity as the square root of the chord of curvature parallel to it. 75. If a body describe a circle by a force in the circumference, and at the same time the circle revolve about the centre of force in its own plane with an angular velocity varying inversely as the square of the body's distance; prove that the path traced out in fixed space is the spiral of Archimedes, and find the law of force by which it may be described. 76. Required the law of force when the space due externally to the velocity in a circle : space due internally :: ve:1; e being the base of the hyperbolic system of logarithms. 77. If any number of hyperbolas have a common centre, and at distances proportional to their major axes double ordinates be drawn ; shew that bodies acted upon by the same absolute force situated in the centre will describe any of the arcs thus cut off in equal times, 78. Prove that the force by which a body may describe any of the conic sections, round a centre of force in the vertex, varies inversely as the square of the distance, and directly as the cube of the secant of the angle which it makes with the axis. 79. A body projected in a given direction with a given velocity and attracted towards a given centre of force, has its velocity at every point : the velocity in a circle at the same distance::1:72; find the orbit described, the position of its apse, the magnitude of its axis and the law of force. 80. If a body describe an equilateral hyperbola, round |