DYNAMICS. CHAPTER I. 1. THE problems usually discussed under this head are those which relate to the geometrical connections between given motions, or given kinds of motion, and those which relate to the action of forces, and the motions and changes of motion produced by forces. The former belong to pure science, and deal with the geometry of motion, a branch of mathematics to which the name Kinematics was applied by Ampère. We shall however employ the word Phoronomy to represent the purely geometrical science of motion in the abstract. Strictly speaking the word Dynamics includes Statics, the discussion of the equilibrium or balancing of forces, and Kinetics, the discussion of the effects of forces on the motion of bodies. Mechanism, including such problems as result from considering trains of wheel-work or any connected machinery, is really a branch of Phoronomy. Some writers employ the word Kinematics to represent what is commonly called Mechanism. To Kinetics belong the consideration of the forces setting such machinery in motion, or keeping it in motion, the problems of Physical Astronomy, and others of important practical application. We shall commence by a development of the formulæ of Phoronomy, and afterwards proceed to consider the applica B. D. 1 tion of the formulæ, and of the Laws of Motion, or Laws of Force, to the determination of the motion of a particle, and of a system of particles, produced by the action of given forces, or, conversely, of the forces required to produce given motions. The idea of a particle, or of a material point, capable of being set in motion, or of having its motion affected, by the action of force, is a mathematical abstraction leading to the simplest forms of Kinetics. The determination of the motions of the bodies constituting the Solar System belongs to this class in virtue of the facts that the Planetary Bodies are nearly spherical in form, and that their dimensions are very small in comparison with their distances from each other and from the Sun. Moreover the mathematical idea of a solid body is that of a system of particles, and the discussion of the motion of a single particle therefore naturally precedes the discussion of the motion of a body or system of particles. It will be seen that Newton's Laws of Motion connect the action of a force on a particle with the accelerations produced, and lead to the formation of differential equations, the integration of which gives the solution of the problem of determining the motion. It will appear further that Newton's Laws are sufficient for the determination of the motion of a system of particles or bodies, whether rigidly connected or not, and lead, in a similar manner, to systems of differential equations containing in their solution the motions of the body, or of the various bodies of the system. CHAPTER II. DIFFERENTIAL EQUATIONS. 2. THERE are certain differential equations which occur so frequently in the discussion of questions in Kinetics, that we think it worth while, for convenience of reference, to give a brief solution of them. dx (1) The equation, dy + yf'(x) = F′(x), is at once solved by the integrating factor |