of descent, prove that the whole space described by the body before the motion ceases KT e k2 4a - 1 where 2k = resistance to velocity 1, g = gravity, 2a = diameter of the generating circle, and 7 = the semi-circumference of a circle the radius of which is 1. 138. A bent lever, of which the arms are a and b, and the angle e, makes small oscillations in its own plane; the length of the isochronous simple pendulum is 2 a3 + 63 3 Vat +64 + 2a2 b2 cos 0 139. State and explain D'Alembert's principle, and apply it to determine the pressure on the axis about which a body revolves when acted on by a single force in a plane perpendicular to the axis. 140. Give Newton's construction for determining the path of a projectile acted upon by gravity in a medium whose resistance = a velocity, and apply the differential equations of motion to determine the actual equation. 141. When any number of bodies move uniformly in straight lines in different planes, their centre of gravity also moves uniformly and in a straight line. 142. Having given the moment of inertia round any axis passing through the centre of gravity of a body, to determine that round any axis parallel to the former. 143. A globe is projected vertically upwards with a given velocity c, in a medium where the resistance is = k x (vel.)?, and is acted on by gravity ; determine the relation between the time, space, and velocity. 1829 144. A pendulum is taken to the top of a hill; how many seconds a day does it lose ? 145. Find the general equation for the motion of a vibrating cord. 146. Find the velocity acquired by a cylinder unrolling and descending vertically through a given space. 147. If two imperfectly elastic bodies impinge obliquely on each other with given velocities and directions, find the velocities and directions of their motions after impact. 148. If a body acted on by gravity be projected from a given point A with a given velocity so as to strike a given point Q, find the direction of projection; and if AI bisects the angle which AQ makes with the vertical, shew that there are two such directions equally inclined to AI. 149. A body urged towards a plane by a force varying as the perpendicular distance froin it, is projected at right angles to the plane from a given point in it with a given velocity. Find what force must act at the same time on the body parallel to the plane, that it may move in a given parabola having its axis in the plane; and determine the circumstances of the motion. 150. A uniform rod is oscillating about one extremity; find the tendency of the ris inertiæ in any given position to bend the rod at any point, and determine at what point that tendency is the greatest. 151. A body is oscillating in a cycloid, in a medium where the resistance varies as the (vel.)?, and the density varies inversely as the arc measured from the lowest point; prove by a method similar to Newton's in Book II. Prop. 26, that the time of descent to the lowest point will be the same from all altitudes; and apply the integral calculus to find the whole time, supposing the resistance at the highest point corresponding to any velocity v, to be less than T' where l is the length of the pendulum. 152. Find the limit of the velocity communicated by a body A to C, through an indefinite number of mean proportionals between A and C, the bodies being supposed perfectly elastic. 22 2 153. Explain D'Alembert's principle, and apply it to determine the motion of two bodies connected together by a wheel and axle; the inertia of the machine being taken into account. 154. The heights of the ridge and eaves of a house are H and h, and the roof is inclined at 30° to the horizon. Find where a sphere rolling down the roof from the ridge will strike the ground, and also the time of descent from the eaves. 155. A river, of which the breadth is a, flows with a velocity u, and a swimmer, whose velocity is nu, always aims at a mark on the farther bank directly opposite to the place where he entered the river. Find the curve in which he swims, and shew that the time of his arriving at the mark is equal to na (n2 – 1) u 156. A uniform rod, of which the elasticity is e, falls upon a smooth horizontal plane : given the altitude from which it falls, and its inclination to the horizon; find its motion after rebounding 157. An imperfectly elastic body slides down a smooth plane of given length, and is reflected from the horizontal plane. Find the inclination of the plane that the range may be a maximum, and find the range. 158. Find the ratio of the height to the diameter of the base of a cylinder, that the moment of inertia may be the same about any axis whatever passing through its centre of gravity. 159. If a body oscillate in a medium in which the resistance varies as the square of the velocity; the differences between the times of oscillation in the medium and in vacuo are proportional to the arcs nearly. (Newton, Book II. Prop. 27.) 160. When a body is acted upon by forces X and Y in the directions of the coordinates x and y, d22 dt2 161. Assuming the ordinary expansion of 8/Vdx, determine the requisite addition to be made to it when V involves the limiting values of a', y, p, ....; and apply the method to find prove dt = X and = Y. the position of the curve of quickest descent from one curve to another, when the motion commences from the first curve. 162. A body moveable about a fixed axis is acted upon by a single force in a plane perpendicular to the axis; find the pressure on the axis arising from that force, and thence determine fully the coordinates of the centre of percussion. 163. When a body moves upon a surface of revolution, find the re-action of the surface. 164. If a body be acted upon by any forces, the motion of the centre of gravity will be the same as if all the forces were applied at that point; and the motion of rotation will be the same as if the centre of gravity were fixed and the same forces applied. 165. Define the radius of gyration of any body or system of bodies moveable about a fixed axis, and investigate an expression for determining its magnitude: apply also this expression to a sphere revolving about a diameter. 166. Find the angles which the axis of instantaneous rotation makes with the coordinates x, y, z. 167. If a point move through one or more spaces bounded by parallel planes, and be acted upon by a force which is perpendicular to the planes, and which is the same at the same distance from them, the angle of incidence is to the angle of emergence in a given ratio. (Neuton, Book I. Prop. 94.) 168. Explain the nature and use of the ballistic pendulum, and perform the requisite calculations in the experiment. 169. If a body move in a surface of revolution acted upon by a centre of force situated in the axis, the areas described, projected on a plane perpendicular to the axis, are proportional to the times. (Newton, Book I. Prop. 55.) 170. Shew that when a body moves in an inverted cycloid, the force by which it is urged along the curve varies as the arc to the lowest point; and hence shew that the oscillations are isochronous. 171. Two bodies whose common elasticity is e, moving with given velocities, impinge directly upon each other ; it is required to determine their velocities after impact. 1830 172. A perfectly smooth rod in a vertical plane revolves uniformly round a vertical axis, and a ring placed on it is attracted to a horizontal plane by a force varying as the distance in addition to the uniform force of gravity ; required the form of the rod that the ring may remain on whatever point it is placed. 173. The equation to the path of a projectile is y = ax + a hyp. log (1 – bx), gravity (= g) acting parallel to the axis of y; shew that the resistance = k velocity. 174. A circular sector revolves through any angle round one of its extreme radii; find the centre of gravity of the solid generated, its density varying as the nth power of the distance from the centre of the circle. 175. In the above case, supposing the angle of the sector and the angle through which it has revolved to remain the same, prove that as the radius varies, the motion of the centre of gravity will be in a plane passing through the centre of the circle; find the line of motion and the equation to the plane. 176. A body acted on by gravity oscillates in a curve, and a chain of given length, suspended from the horizontal ordinate where the motion commences, is divided by the ordinate at each point in two parts proportional to the two parts of the tension at that point arising from the centrifugal force and from gravity. What is the curve? 177. A paraboloid revolving round its axis strikes a body P in a direction perpendicular to the radius, and P, being attracted to the intersection of the radius and axis by a force varying 1 after impact describes a parabola of the same dimensions D2' as the generating one. Determine the velocity of rotation and the point of impact. 178. If a pendulum of length 1 vibrate in a small circular arc in a medium of which the resistance = kv2 to velocity v, and if s be the arc described from the commencement of a vibration to the point where the velocity is greatest when the as |