spectively; it is required to find a point C, in the quadrant AB, to which if a tangent be drawn meeting the radius OB produced in D and a line touching the circle at A in the point E; so that the time down DE plus the time of moving along EA with the acquired velocity may be equal to time down the vertical diameter. 68. Two bodies, A and B, descend from the same extremity of the vertical diameter of a circle, one down the diameter, and the other down the chord of 30°. Find the ratio of A to B when their centre of gravity moves along the chord of 120°. 69. Two circles are situated in the same vertical plane; determine analytically and geometrically the straight line of quickest descent from one to the other; and show that the two results agree. 70. The plane of a circle is inclined at any given angle to the horizon; show that the times of descent down any chord from the highest point are the same. 2' 71. A body, whose elasticity =, projected from the floor of a room 12 feet high strikes the ceiling and floor and just reaches the ceiling again; find the velocity of projection, 72. A rocket ascending vertically with a velocity of 100 feet in I" explodes when it has reached its greatest height, and the interval between the sound of the explosion reaching the place of starting, and a place-mile distant, is 1". Determine the velocity of sound. 4 73. A right-angled triangle being placed with its two sides horizontal and vertical respectively, it is required to determine their proportion so that the time of falling down the perpendicular and describing the base with the velocity acquired may be equal to the time of descent down the hypothenuse. 74. Find the straight line of quickest descent from a given point within a circle to the circumference. 75. Find the straight line of quickest descent from a given circle to a given circle within it. 76. Find the straight line of quickest descent from a given circle to a given point, (1) within the circle, (2) without it. 77. Find the straight line of quickest descent from the focus to the curve of a parabola whose axis is vertical. 78. A ball of given elasticity is projected vertically upwards with a velocity of 40 feet in I"; it returns to the point of projection, which is on a hard horizontal plane, and rebounds; it returns again and rebounds, and so on, till the motion ceases; required the whole space described. 79. Two balls, whose weights are 9 lbs. and 2 lbs. respectively, are connected by a string 15 feet in length; the former weight rests on a smooth horizontal table along which it is drawn by the latter that begins to fall vertically; find the velocity required after falling through (1) 12 feet, and (2) 20 feet, and the times of motion in both cases. 80. A weight P, after falling freely through a feet, begins to raise up a weight Q connected with it by means of a string passing over a fixed pulley; required the subsequent motion. 81. A mass of 18 lbs. is so distributed at the extremities of a cord passing over a fixed pulley that the more loaded end descends through 13 feet in as many minutes; required the weights at each end. 82. Two equal weights W are suspended over a fixed pulley; what weight must be added to one of them that it may descend through 100 feet in 8"? 83. A weight of 7 lbs. draws up one of 5 lbs. over a fixed pulley; at the instant of letting go the weight of 7 lbs. a velocity 4 feet in I" is communicated to it; how far will it descend in 8", and what velocity will it have acquired at the end of that time? 84. The accelerating force on the centre of gravity of two bodies, P and Q, moving vertically and connected by a string passing over a fixed pulley = P+Q .g. 85. The major axis of an ellipse is vertical; determine the radius vector measured from the upper focus, down which the time of descent is the least possible. ON MOTION UPON A CURVE AND ON THE SIMPLE PENDULUM. General Formula. If denote the number of inches in the length of a simple pendulum, t" the time of one oscillation, g the measure of the force of gravity (=32.2 feet generally), then The length of a second's pendulum in London =39*1393 inches. 86. Three planes A, B, C are in contact; A is vertical, B and C are inclined to the horizon at angles 60° and 30° respectively; find the velocity with which a body beginning to descend from A will begin to move along the horizontal plane passing through the lower extremity of the plane C. 87. If an inelastic body move uniformly along one side of a regular polygon, show that it will continue to describe the other sides uniformly, but with velocities decreasing in geometrical progression; and in the case of a hexagon, show that the time of describing the first side time of describing the last :: I: 32. 88. Having given the length of a pendulum that will oscillate seconds; find the length of a pendulum that will oscillate 4 times in I"; and another 9 times in 1'. 89. Find the length of a pendulum which oscillates as often in I' as there are inches in its length. 90. Two pendulums, the lengths of which are L and 7, begin to oscillate together, and are again coincident after n oscillations of L. Given L to find l. 91. Find the time of an oscillation of a pendulum 11 feet in length 3 miles above the earth's surface at the equator, where the second's pendulum equals 38.997 inches. 92. A second's pendulum is carried to the top of a mountain and there loses 48"-6 in a day; determine the height of the mountain, supposing the earth's radius to be 4000 miles. 93. A pendulum which should beat seconds is found to lose 10" a day. Determine the quantity by which its length should be increased or diminished. I (dist.)2 94. The force of gravity varying as from the centre of the earth, how high must a second's pendulum be carried above the level of the sea that it may vibrate 59 times in 1', the radius of the earth being 3958 miles? 95. The times of oscillation of a pendulum are observed at the earth's surface and at a given depth below the surface; find from these data the radius of the earth which is supposed to be spherical. 96. The length of a pendulum that vibrates sidereal seconds being 38.926 inches; find the length of a sidereal day. Find the increment of the length of the pendulum that it may measure mean solar time. 97. The length of a second's pendulum being 39.06 inches at the equator, 39.28 at the pole, and 39.2 in latitude 52°; find the force of gravity at the equator and at the pole. 98. A second's pendulum is lengthened 1.05 inches; find the number of seconds it will lose in 12 hours. 99. Prove that the times of vibration of the same pendulum when carried to different heights above the earth's surface are to each other as the distances of those heights from the earth's centre. 100. A pendulum gains o"-05 in an hour before it is carried up a high mountain; at what height in the ascent will the pendulum keep true time? ON THE MOTION OF PROJECTILES IN A NON- General formulæ. If two straight lines be drawn through the point of projection, one horizontal, the other vertical; and these lines be taken for the coordinate axes of x and y respectively; V the velocity of projection, and a the angle which the direction of projection makes with the axis of r; h the impetus which equals V2 the space due to the velocity, and hence h= where g mea 2g sures the force of gravity; then the equation to the curve described by the projectile is 1. y=x tan a 9x2 2V2 cos2α or r tan a x2 From this equation most of the properties respecting projectiles may be deduced. If R be the horizontal range and T" the time of describing this range, H the greatest height, then sin a, 3. R=2h sin 2 a. 4. H=h sin1a. If w be the weight of a ball or shell, p the weight of the gunpowder used in firing the ball or shell from a mortar, and v the velocity generated by the powder, then 101. If a body be projected with a velocity of 850 feet in 1" in a direction making an angle of 60° with the horizon; find the focus of the parabola described, and also its latus rectum. 102. A body is projected in a direction making an angle of 15° with the horizon, with a velocity of 60 feet in 1"; find its range, greatest altitude, and time of flight. 103. A body is projected at an angle of 45°, and descends to the horizon at a distance of 500 feet from the point of projection; with what velocity was it projected, what was its greatest altitude, and the whole time of flight? 104. Find the velocity and direction of projection of a ball that it may be 100 feet above the ground at one mile distance, and may strike the ground at two and a half miles. 105. Compare the space described by a projectile in the direction of projection with its vertical fall in the same time. If the velocity be given, determine the angle of projection that the focus may lie in the horizontal line through the point of projection. 106. The horizontal range of a projectile is 1000 feet and the time of flight is 15"; find the direction and velocity of projection; also the greatest altitude of the body during the flight. 107. If the horizontal range of a body projected with a given velocity be three times the greatest altitude, find the angle of projection. What is the value of this angle when the range is equal to the altitude? 108. A shot is fired with a given velocity towards a tower whose horizontal distance from the cannon is one-half the range, and whose altitude subtends an angle tan13 at the point of projec = 4 tion; find the inclination of cannon to the horizon that the shot may strike the summit of the tower. 109. From one extremity of the base (500 feet) of an isosceles triangle, whose vertical angle is 36°, situate in a vertical plane, a body is projected in the direction of the side adjacent to that extremity so as to strike a body placed in the other extremity; find the velocity of projection, and the time of flight. 110. A shell being discharged at an angle of 45°, the sound of its explosion was heard at the mortar 3"-5 after the discharge; required the horizontal range, the velocity of sound being 35 g in 1". III. A body is projected at an angle of 60° elevation with a velocity of 150 feet in 1"; find the direction and velocity of the projectile after the lapse of 5"; and its height above the horizontal plane passing through the point of projection. 112. Two bodies are projected from the same point with the same velocity; the directions of projection are measured by the angles a and 2a respectively; compare the areas of the parabolas described, supposing the horizontal ranges equal. 113. If the areas in the last problem be equal; what is the value of a? 114. A body is projected with a given velocity from a given point and in a given direction; find where it will strike a given plane. 115. A body is projected from a given point with a given velocity; to find the direction that it may just touch a given plane. 116. A body is projected from the summit of a hill, whose form is a right cone the vertical angle of which is 120°, in a given direction with a given velocity; to find where the projectile will strike the hill. 117. A body is projected from the top of a tower 200 feet high with a velocity of 50 feet in 1" and at an angle of elevation =60°; find the range on the horizontal plane passing through the foot of the tower, and the time of flight. 118. From the top of a tower two bodies are projected with the same velocity at different given angles of elevation, and they strike the horizon at the same place; to find the height of the tower. 119. Show that a body projected in an oblique direction along an inclined plane describes a parabola, and find its latus rectum, having given the inclination of the plane, also the velocity and direction of projection. 120. A body projected in a direction making an angle of 30° with a plane whose inclination to the horizon is 45°, fell upon the plane at the distance of 250 feet from the point of projection, which L |