2 both values of λ; let its values be n, and n2, and we have, k1, 1, k2, 2, being the corresponding values of k and 4, where a1, a2, B1, B2, are arbitrary constants. 19 1 = 2 Having given the initial values of 0, 0', " dt find a,,a,, B1, B2, and thus the solution is complete. It may be noticed that the values of 0 and may be found at any time by taking the algebraic sum of the corresponding values of the inclinations to the vertical of two pendulums whose times of oscillation are and 2π 2π n2 Also, if be com mensurable, the system will in time return to its first position, and the motion will be periodic. A very slight modification of the process gives us the result of small displacements not in one plane: but the student may easily work out these for himself. We have here a simple example of the principle of the "Coexistence of Small Oscillations;" but this, for its satisfactory treatment in the general case, requires the use of D'Alembert's Principle; which, though (§ 69) merely a corollary to the Third Law of Motion, and as such clearly pointed out by Newton, is beyond the professed limits of the present treatise. 322. The examples, which have just been given, may suffice to convey an idea of the mode of applying our methods to any proposed case of motion of two constrained particles. These methods are applicable to more complicated cases, when more particles than two are involved; but nothing would be gained by such a proceeding, as D'Alembert's Principle supplies us with a far simpler mode of investigating the motions of any system of free or connected particles: especially when it is simplified in its application by the beautiful system of Generalized Co-ordinates introduced by Lagrange. See Thomson and Tait's Natural Philosophy, § 329. EXAMPLES. (1) Two spheres whose masses are M and M' are placed in contact, and one of them is projected in the line of centers with velocity V. If the law of attraction be D2, find where, and after what time, they will meet. (2) If the sun were broken up into an indefinite number of fragments, uniformly filling the sphere of which the earth's orbit is a great circle, shew that each would revolve in a year. (3) A thin spherical shell of mass M is driven out symmetrically by an internal explosion. Shew that if, when the shell has a radius a, the outward velocity of each particle be v, the fragments can never be collected by their mutual attraction unless M 22< a (4) Two equal particles are initially at rest in two smooth tubes at right angles to each other. Shew that whatever were their positions, and whatever their law of attraction, they will reach the intersection of the tubes together. (5) In last question suppose the original distances from the intersection of the tubes to be a, b, and the attraction as the square of the distance inversely, find the future paths if at any instant the constraint is removed. (6) A number of equal particles, attracting each other directly as the distance, are constrained to move in parallel tubes; if the positions of the particles be given at the commencement of the motion, determine the subsequent motion of each; and shew that the particles will oscillate symmetrically with respect to the plane perpendicular to the tubes which passed through their center of inertia at the commencement of the motion. (7) Two given masses are connected by a slightly elastic string, and projected so as to whirl round, find the time of a small oscillation in the length of the string. 1 Give a numerical result, supposing the masses to weigh 1lb. and 2 lbs. respectively, and the natural length of the string to be 1 yard, and supposing that it stretches th inch for a tension of 1 lb. 10 (8) Two equal masses M, are connected by a string which passes through a hole in a smooth horizontal plane. One of them hanging vertically, shew that the other describes on the plane a curve whose differential equation is (9) Two equal particles connected by a string are placed in a circular tube. In the circumference is a center of force 1 D One particle is initially at its greatest distance from the center of force, shew that if v, v' be the velocities with which they pass through a point 90° from the center of force, 1 (10) Two equal balls repelling each other with a force hang from the same point by strings of length 7. Shew D2 that if when in equilibrium, the strings making an angle 2a with each other, they be approximated by equal small arcs, the time of an oscillation is the same as that of a pendulum whose length is (11) One of two equal particles connected by an inelastic string moves in a straight groove. The other is projected parallel to the groove, the string being stretched; determine the motion, and shew that the greatest tension is four times the least. (12) Two particles are connected by an elastic string of length 2a, and one is projected perpendicularly to the string when it is unstretched. Shew that in the relative orbit (13) Two equal particles connected by a rigid rod move on a vertical circle. If they be slightly disturbed from the higher position of equilibrium, determine the motion. Also find the time of a small oscillation about the position of stable equilibrium. (14) Two particles P and Q are connected by a rigid rod. P is constrained to move in a smooth horizontal groove. If the particles be initially at rest, PQ making a given angle with the groove in a vertical plane through it, find the velocity of when it reaches the groove, and shew that Q's path in the vertical plane is an ellipse. (15) A particle of mass m has attached to it two equal weights m' by means of strings passing over pulleys in the same horizontal plane, and is initially at rest halfway between them. Determine the motion. Shew that if the distance between the pulleys be 2a, the velocity of m will be zero when it has fallen through a space and that it will have its greatest velocity when passing through its position of statical equilibrium. (§ 73.) (16) Two masses M, M' are connected by a string which passes over a smooth peg. To M' is attached a string which supports a mass m such that M'+m= M, and m is displaced through an angle a. Investigate the motion, supposing m so small that the horizontal motion of M' may be neglected. Shew that the string M'm will be vertical after the time (1) A spiral spring is stretched an inch by each additional pound appended to its lower end; find the greatest velocity which will be acquired by a mass of 20 lbs. appended to the unstretched spring and allowed to fall. Also find how far the mass will fall, and the time of a complete oscillation. (2) Find the form of the hodograph, and the law of its description, for any point of one circular disc rolling uniformly on another. Hence, find the force under which a free particle will describe an epitrochoid, as it is described by a point of the uniformly rolling disc. (3) Form the equation to the surfaces of equal time, as those of equal action were found in § 254. (4) Apply a method similar to that of § 255 to find the equation to the common brachistochrone. (5) Find the law of the force when the brachistochrone is an ellipse with the center of force in its focus. (6) A rod slides between two rough parallel horizontal bars, in a plane perpendicular to the bars: determine the |