12. Rectilinear motion. 13. General motion of a particle. 14. Central forces. Hodograph. 15. Kinetics of a system of discrete particles. 16. Kinetics of a rigid body. Fundamental principles. 17. Two-dimensional problems. 18. Equations of motion in three dimensions. 19. Free motion of a solid. 20. Motion of a solid of revolution. 21. Moving axes of reference. 22. Equations of motion in generalized co-ordinates. 23. Stability of equilibrium. Theory of vibrations. PART I.--STATICS § 1. _Statics of a Particle._--By a _particle_ is meant a body whose position can for the purpose in hand be sufficiently specified by a mathematical point. It need not be "infinitely small," or even small compared with ordinary standards; thus in astronomy such vast bodies as the sun, the earth, and the other planets can for many purposes be treated merely as points endowed with mass. A _force_ is conceived as an effort having a certain direction and a certain magnitude. It is therefore adequately represented, for mathematical purposes, by a straight line AB drawn in the direction in question, of length proportional (on any convenient scale) to the magnitude of the force. In other words, a force is mathematically of the nature of a "vector" (see VECTOR ANALYSIS, QUATERNIONS). In most questions of pure statics we are concerned only with the _ratios_ of the various forces which enter into the problem, so that it is indifferent what _unit_ of force is adopted. For many purposes a gravitational system of measurement is most natural; thus we speak of a force of so many pounds or so many kilogrammes. The "absolute" system of measurement will be referred to below in PART II., KINETICS. It is to be remembered that all "force" is of the nature of a push or a pull, and that according to the accepted terminology of modern mechanics such phrases as "force of inertia," "accelerating force," "moving force," once classical, are proscribed. This rigorous limitation of the meaning of the word is of comparatively recent origin, and it is perhaps to be regretted that some more technical term has not been devised, but the convention must now be regarded as established. [Illustration: FIG. 1.] The fundamental postulate of this part of our subject is that the two forces acting on a particle may be compounded by the "parallelogram rule." Thus, if the two forces P,Q be represented by the lines OA, OB, they can be replaced by a single force R represented by the diagonal OC of the parallelogram determined by OA, OB. This is of course a physical assumption whose propriety is justified solely by experience. We shall see later that it is implied in Newton's statement of his Second Law of motion. In modern language, forces are compounded by "vector-addition"; thus, if we draw in succession vectors [->HK], [->KL] to represent P, Q, the force R is represented by the vector [->HL] which is the "geometric sum" of [->HK], [->KL].