90. Compare the time in which a sphere slides dowu an inclined plane with the time in which it rolls down the same plane. 91. A given uniform rod moves in the same plane in a hemisphere. Determine its motion. 92. Several bodies are projected from a given point, with the same velocity, in different directions, being acted upon by the force of gravity. Find the locus of them all at the end of a given time. 93. Find the elasticity of two bodies A and B, and their proportion to each other, so that when A impinges upon B at rest, A may remain at rest after impact, and B move on with an nth part of A's velocity. 94. If a solid cylinder and a thin hollow cylinder of the same weight and radius roll together from rest down a given inclined plane, how far will they be separated after a given time? 95. Two bodies are projected from the same point in a horizontal plane with equal velocities and have the same horizontal range. Required the directions of projection so that the area included between the parabolas described may be the greatest possible. 96. Find the time in which a pendulum would oscillate in a hypocycloid within the earth, the diameter of the wheel being half the earth's radius. 97. P descends drawing Q over a fixed pulley. Find the space described in a given time, the string being conceived to have weight. 98. A straight bar of given length is made to oscillate in its own plane about an axis situated in a line which bisects it at right angles. Required the point of suspension, so that the time of oscillation may be the least possible. 99. Find the curve described by a body projected in a medium, the resistance of which varies as the velocity, and acted upon by gravity. 100. Prove that an arc of a circle, which does not exceed 60°, is a curve of quicker descent than any other curve which can be drawn within the same arc: and that the arc of 90° is a curve of quicker descent than any other curve which can be drawn without the same arc. 101. Find the time of a small oscillation of an oblique-angled 1826 parallelogram vibrating in its own plane about an axis passing through one of its angular points. 102. If from a point in a horizontal plane, any number of bodies be projected in the same vertical plane with such velocities and in such directions, that the areas of the parabolas described shall be equal to one another ; find the curve which shall touch them all. 103. Two circles are situated in the same vertical plane; determine analytically and geometrically the straight line of swiftest descent from one to the other; and shew that the two results agree. 104. Find the moment of inertia of a rectangle revolving in its own plane, round an axis passing through its centre of gravity. 105. From the top of a tower two bodies are projected with the same given, velocity at different given angles of elevation, and they strike the horizon at the same place. Find the height of the tower. 106. Two spheres of given magnitudes and elasticity, not affected by gravity, are projected at the same time from given points with given velocities in opposite directions in the same straight line ; find when and where their impact takes place, and their positions at the end of any assigned time after impact. 107. If the force be repulsive and vary as the distance from the centre of the globe; prove that the oscillations in an epicycloid are isochronous; and having given the radii of the globe and wheel, find the velocity at any point and the actual time of an oscillation. 108. Find the motions of two equal balls connected by an inflexible rod without weight, one of them being attached to a N given weight by means of a string passing over a fixed pulley, and the other moving on a perfectly smooth horizontal plane. 109. A straight rod which is always parallel to the horizon descends freely by the force of gravity, and at the same time revolves uniformly about one of its extremities; required the equations to the surface traced out by it, and to the tangent plane at any point. 110. A body is projected perpendicularly upwards, and the time between its leaving a given point and returning to it again is given; find the velocity of projection and the whole time of motion. 111. A globe of given weight and radius rolls down the surface of a hemispherical bowl from rest; find the velocity acquired at any point of its descent. 112. The bias and velocity of projection of a bowl are such as to cause it to describe a given logarithmic spiral; find the direction of projection, so that, after having described a path of given length, it may impel the jack in a given direction. 113. A body projected in the direction of the action of a constant force, describes P and Q feet in the pik and oth seconds; find the magnitude of the force and the velocity of projection. 114. Two bodies begin to descend at the same time down two given inclined planes from given points in the same vertical line; find their distance from each other at the end of any assigned time. 115. An uniform rod is made to vibrate about a point, so that the time of its oscillation is a minimum; find the force exerted on the point of suspension in any given position. 116. A semi-circle, the plane of which is vertical and base horizontal, has an uniform chain of given length, placed in a given position upon its circumference; find the velocity of the chain at the end of a given time. 117. Two given weights are connected by a string passing through a hole in a horizontal plane: one of them is projected in any direction in the horizontal plane, the other descends vertically by the action of gravity ; find the motion of the bodies, and the curve described on the plane. 118. If a body be projected in a medium, the resistance of which varies as the velocity, and be acted on by gravity, and another be projected in vacuo at the same angle, and with the same velocity, and acted upon by the same constant force, and if t, and to be the times of describing two arcs in the medium, and in vacuo, so related to each other that the tangents at their extremities shall cut the axis at the same angle; the ekt, - 1 kt, ; k being the resistance to velocity 1. 119. If a body be projected downwards with a given velo- 1827 city, what is its velocity after describing a given space ? 120. Prove, that when a mass entirely free is struck at any point, the motion of translation is the same as if the direction of the impact passed through the centre of gravity, and that of rotation as if the centre were fixed by an axis. By this proposition find the distance of the centre of percussion from a fixed axis. 121. Find, geometrically, the inclination of the path of a projectile to the horizon at a given time after the beginning of its motion, and prove by that means that the trajectory is a parabola. 122. Explain the division of a string which produces the several notes of the diatonic scale. What alteration would be made in the general pitch by assuming 256 instead of 240 for the number of vibrations constituting the tenor C ? 123. If two elastic balls in the ratio of 1 to 3 meet directly with equal velocities, the larger one will remain at rest. 124. Shew that a tennis ball projected along an inclined roof, but not in the direction in which it would naturally fall, describes a parabola, and find its latus rectum, having given the inclination of the roof, and the velocity and direction of projection. 125. If a circle of given radius oscillate flatways through a small angle, determine the content of the solid which it traces out, having given the time of the oscillation and the whole angle through which it oscillates. 126. Find how long a given sphere, suspended by a twisted string which is suffered to untwist, will continue to turn in the same direction. 1828 127. By what experiments is the third law of motion established ? 128. Two bodies of given magnitudes and elasticity impinge directly upon each other with given velocities ; find the velocity of each after impact. 129. A body is projected from a given point in a given direction with a given velocity, and acted upon by gravity ; find where it will strike a given plane. 130. Find the velocity and direction of projection of a hall, that it may be 100 feet above the ground at one mile distance, and may strike the ground at three miles. 131. A straight rod moves on a smooth horizontal plane, subject to the condition of always passing through a given point: determine its motion. Prove that the varying centre of gyration of the rod with respect to the fixed point will describe areas uniformly about that point. 132. Find the time of a body's descent down any arc of a cycloid, and shew that the times of the whole oscillations are as the square roots of the lengths of the strings. 133. If a rigid body oscillate about a horizontal axis, find the length of a simple pendulum which shall oscillate in the same time. 134. A uniform rod is at liberty to move freely in a vertical plane about a horizontal axis; find the nature of the circumference of a wheel which, revolving uniformly about a given horizontal axis, shall cause the rod to revolve uniformly also: the point of contact of the wheel and the rod being always at the same distance from the point of suspension. 135. A ring slides down a perfectly smooth rod revolving uniformly in a vertical plane ; find the motion of the ring. 136. An oblique parallelopiped oscillates about one of its edges which is in a horizontal position; determine its motion and the pressure it exerts against the axis of suspension in any position. 137. If a body oscillate in a cycloid, in a medium the resistance of which varies as the velocity, and s be the first arc |