36. The resultant and sum of two forces are given, and also 1826 the angle which one of them makes with the resultant; it is required to determine the forces and the angle at which they act.

37. If t be the length of a part of the catenary, the weight of which is equal to the tension at a point whose abscissa is x and corresponding arc s, prove that

38. A chain suspended at its extremities from two tacks in the same horizontal line forms itself into a cycloid; prove that the density at any point ∞ sec3 (0), and the weight of the corresponding arc o tan (0), being the arc of the generating circle measured from the vertex.

39. One end of a beam is connected to a horizontal plane by a hinge, about which the beam is suffered to revolve in a vertical plane; the other end is attached to a weight by means of a string passing over a pulley in the same vertical plane; find the position of equilibrium.

40. Find the pressure which a given power exerts by means 1827 of a common vice, the dimensions of which are given.

41. The voussoirs of a bridge being very small, and the equilibrium maintained by the vertical pressure of the masonry above them, of what portion of a circle must the intrados consist, that the ascent of the bridge from a level road may be continuous, supposing the thickness of the arch at its summit to be to the radius of the intrados as 1 to 5?

42. An uniform elastic string being of such a length that, when it hangs vertically, if an equal quantity were appended to the lowest point it would stretch it to twice that length, what weight must be appended at the middle point that the increase of length may be three quarters of the original?

43. Two given weights being attached to given points in the circumference of a wheel, find the position in which the greatest weight will be supported on the axle.

44. If two given equal weights sustain each other by a string passing over a smooth curve, the plane of which is vertical, the sum of the pressures on any arc depends only on the directions of its extremities.

1828

45. If the particles of a hollow elastic cylinder be so arranged, that on its being subjected to a given internal pressure they may all be in the same given degree of dilatation, find how the thickness must be altered, in order that the strength of the cylinder may increase in arithmetical progression; the internal radius of the cylinder being supposed to remain constant.

46. If a point be kept at rest by three forces acting upon it at the same time, any three lines which are in the direction of those forces and form a triangle will represent them.

47. A body is placed on a horizontal plane; find when it will be supported.

48. A ladder of uniform thickness rests with its lower end on a horizontal plane, and its upper end on a slope inclined 60° to the horizon: the ladder makes an angle of 30° with the horizon; find the force which must act horizontally at the foot to prevent sliding.

49. In a given sphere rests a given plane triangle of uniform thickness; find the angle which it makes with the horizon.

50. Find the form of a uniform chain suspended from any two points on the surface of an upright cone, and resting on the curve surface. Find the tension when it becomes a hori

zontal circle.

51. The equilibrium of the screw will take place when the power is to the weight as the distance of two contiguous threads to the whole circle described by the point where the force is applied.

52. If any number of forces act in the same plane upon a rigid body, determine their resultant, and the equation of the straight line in which the resultant acts.

53. Prove the formula for the place of the centre of gravity

of any body, viz. h = Sædm, and apply it to find the centre of

m

gravity of a common parabola.

54. State the most recent and approved experiments, whereby it is ascertained that the decrement of velocity arising from friction is the same for all velocities.

55. When any number of forces act on a body, shew that the plane on which the sum of the projections of the moments is a maximum, is perpendicular to the planes with respect to which this sum is 0.

56. If two weights acting perpendicularly upon a straight 1829 lever on opposite sides of the fulcrum, or two forces in opposite directions on the same side of it, are inversely as their distances from the fulcrum, they will balance each other.

57. If on an isosceles wedge, of which the angle is 2a, a power P acting perpendicular to the base, balance a resistance W acting on each of the sides in a direction making an angle with a perpendicular to the side,

P: W:: sin a: cos.

ι

58. When a system is in equilibrium, if a small motion be given to its parts, the centre of gravity will neither ascend nor descend.

59. A cone and sphere of given weights support each other between two given inclined planes, the cone resting on its base. Determine what must be the vertical angle of the cone, that the equilibrium may subsist.

60. Find the ratio of the power to the weight in that system where each pulley hangs by a separate string; first, when the strings are parallel; secondly, when they are not.

61. Find the resultant of any number of parallel forces acting on a rigid body, and shew that they cannot in all cases be reduced to a single force which shall have the same effect.

62. Determine the equation to the catenary, the force of gravity being supposed constant.

63. A weight W is suspended from a point P of an uniform catenary APA'. O and O' are the lowest points of two uniform catenaries, of which AP and A'P are parts. Shew that W is equal to the difference or sum of the weights of the portions OP, O'P of the catenaries, according as AP and A'P are one or both less than a semi-catenary.

64. The content of any segment of a right or oblique prismatic solid is equal to the area of one end of the segment, multiplied into the perpendicular let fall upon it from the centre of gravity of the area of the other end.

65. If three parallel forces acting at the angular points A, B, C of a plane triangle are respectively proportional to the opposite sides a, b, c; prove that the distance of the centre of parallel forces from A

1830

66. A body being acted upon by any number of forces in the same plane; find the equations of equilibrium.

67. A cord passing round a fixed point is drawn in different directions by two equal forces acting at a given angle; find the pressure on the point.

68. Explain the method of graduating the common steelyard.

69. Find the relation between the power and the weight when there is an equilibrium on the inclined plane.

70. Find the centre of gravity of any system of points what

ever.

71. Find the resultant of any number of forces acting in the same plane upon a rigid body, and the equation to the line in which it acts.

72. Investigate the equation to the catenary between the arc and abscissa; and shew that the tension at the vertex is equal to the weight of a portion of the catenary of the same length as the radius of curvature at the vertex.

73. A uniform rod rests with one of its extremities in a semi-circle whose axis is vertical, find the nature of the line supporting its other extremity so that it may rest in every position.

74. If a hemisphere and paraboloid of equal bases and similar materials have their bases cemented together, the whole solid will rest on a horizontal plane on any point of the spherical

surface if the altitude of the paraboloid

the radius of the hemisphere.

3

= a

a being

75. State the principle of virtual velocities, and prove it when two bodies are in equilibrium on a bent lever.

76. When a chain fixed at two points is acted upon by a central attractive or repulsive force, the tension at any point is inversely as the perpendicular from the centre of force upon the tangent at that point.

77. If two equal weights act perpendicularly on a straight 1831 lever, they may be kept in equilibrium round any fulcrum by the same force as if they were collected at the middle point between them.

78. Define the centre of gravity, and find it in a plane triangle.

79. Two weights keep each other in equilibrium on a bent lever:

(1) Compare them.

(2) Prove that if an indefinitely small motion be given, the centre of gravity will neither ascend or descend.

80. Assuming that if dp, dq, dr be the virtual velocities of three forces P, Q, R which keep a point at rest,

PSp+Q8g + Rdr = 0,

in whatever direction the virtual motion of the point takes place; prove that the forces are proportional to the sides of a triangle drawn in their directions.

81. A ladder rests with its foot on a horizontal plane, and its upper extremity against a vertical wall; having given its length, the place of its centre of gravity, and the ratios of the friction to the pressure both on the plane and on the wall; find its position when in a state bordering upon motion.

82. Find the relation of the power to the weight when there is equilibrium on the screw.

83. A body is supported on a plane curve by forces X and Y acting in the directions of the rectangular axes of a and y; prove that

= 0.