(16) A particle falls to the lowest point of a cycloid down any arc of the curve, the axis of the cycloid being vertical and its vertex downwards: to find the position of the particle when the vertical component of its velocity is greatest. It is greatest when the particle has completed half its vertical descent. (17) Two particles are connected by a fine inextensible string at full stretch in a narrow cycloidal tube, the axis of the cycloid being vertical and the vertex upwards: to find the tension of the string during the motion of the particles. Let m, m', be the masses of the particles, c the length of the string, and 2a the length of the axis of the cycloid: then, throughout the motion, the required tension is equal to (18) Two equal molecules are connected together by a fine inelastic thread: one of them is placed on a smooth table, the other just over the edge, the thread being at full stretch at right angles to the edge: to find the whole interval of time. from the commencement of the motion to the instant when the thread first becomes horizontal. to If c be the length of the string, the required time is equal (19) There are three fixed pegs in a horizontal line, the middle peg being equidistant from the other two: to the outside pegs are attached the ends of a fine inelastic thread, which hangs over the middle peg: beads of equal weights rest on the middle points of the two halves of the thread. Supposing the beads to be slightly displaced, in vertical directions, from their positions of equilibrium without slackening the thread, to find the length of a simple pendulum, isochronous with their subsequent oscillations. Let c be the depth of either bead below the line of the pegs when the system is in a position of equilibrium, and a the corresponding inclination of each portion of the thread to the horizon: then the length of the pendulum is equal to c sec❜a. (20) Two candles of equal weights are at rest in vertical positions, being attached to a perfectly flexible wire of insensible mass, which passes over a smooth pully: supposing one of them to be lighted, and to burn out before reaching the pully, in a time t, to find through what space the other candle will have descended by the end of this time. The required descent is equal to (21) A particle, attracted by a force tending towards a fixed point A, (fig. 138), and varying directly as the distance, describes the arc OP and the chord OP of a fixed smooth curve in the same time, whatever point P be chosen in the curve: the particle has no motion when at 0. To find the nature of the curve. The curve is the Lemniscata, the centre of which is at O and of which the axis is inclined to OA at an angle Bonnet; Liouville, Journal de Mathématiques, Av., 1844. (22) A particle falls under the action of gravity down an arc OB (fig. 139) of one of the loops of a Lemniscata, of which the axis OA is inclined at an angle of 45° to the horizon; to determine the time of the descent.. Let a denote the semi-axis of the corresponding equilateral hyperbola, the angle between the chord OB and the axis OA of the loop, and T the required time. Then This expression for the time is the same as that for the descent of a particle down the chord OB; a mechanical property of the Lemniscata which was discovered by Saladini, Memorie dell' Istituto Nazionale Italiano, Tom. I. parte 2. (23) A spherical particle A impinges with a velocity u in a horizontal direction upon a spherical particle B, which is resting at the lowest point of an inverted cycloid, of which the axis is vertical; to determine the velocities of A and B after any number of impacts, the volumes of the particles being equal, while their masses differ in any proposed degree. The velocities of A, B, after x impacts, will be respectively, e denoting their common elasticity and m, m', their masses, SECT. 2. Pressure of a moving Particle on immoveable plane Curves. The general value of the reaction of a curve against a particle, which is moving along the curve, is given by the formula dy 1 ds2 v2 = N+ ρ ...(A), where N represents the resolved part of the whole accelerating force on the particle estimated along the normal in an opposite direction to that in which the reaction R exerts itself, and p denotes the radius of curvature of the curve. In this formula the positive or the negative sign is to be taken according as the particle is moving on the concave or on the convex side of the curve. This formula was first given by L'Hôpital' in the discussion of John Bernoulli's problem of the Curve of Equal Pressure. When the expression for R becomes equal to zero, the particle 1 Mém. de l'Acad. des Sciences de Paris, 1700, p. 9. will either leave the curve or will move along it freely without experiencing any reaction; and the analytical condition shews that, on the commencement of free motion, the normal accelerating force and the centrifugal force of the particle must be equal and opposite. (1) A particle, starting with a given velocity from the vertex of a parabola, of which the axis is vertical, descends down the convex side of the curve by the action of gravity; to find the reaction of the curve at any point of the descent. The resolved part of the force of gravity along the normal in dy a direction opposite to the reaction is 9 and therefore by (A), the particle moving on the convex side of the curve, R=9 ds Now, the equation to the parabola being y2 = 4mx, Also, if h be the altitude due to the initial velocity of the particle, we have If h=m, then the pressure during the whole motion will be equal to zero; and the particle will describe the parabola freely. If h were greater than m, since, from the nature of the case, R cannot have any negative value, the particle would from the first proceed in a path different from the parabola in question. If, instead of supposing the particle to move on a mere curve, we were to conceive it to be moving within an indefinitely thin parabolic tube, R might be negative; and in fact always would be negative, supposing h to be greater than m, when the motion would be the same as if the particle were moving along the concave side of the parabolic curve. Euler; Mechan. Tom. II. p. 64. (2) A particle, starting from rest, descends down the convex side of a circle from a given point in its circumference; to find where it will leave the curve. Let 0 (fig. 140) be the centre of the circle, AO being a vertical radius. Let P be the initial position of the particle, Q its point of departure; PM, QN, horizontal lines. Join OQ, and let A0Q=; let a = the radius of the circle. Then, the centrifugal force at Q being equal to the normal component of gravity, we have Fontana; Memorie della Societa Italiana, 1782, p. 175. (3) A particle is moving along the convex side of an equiangular spiral, towards the pole of which it is attracted by a force varying as any power of the distance; to determine the reaction of the curve at any time during the motion. Let be the distance of the particle from the pole at any time, ur" the attractive force, a the constant angle between the curve and the radius vector, ẞ the initial velocity, and a the initial value of r. Then, by the formula (A), N being equal to |