of kinematical ideas, it is easy to base the whole science of dynamics on these principles ; and there is no necessity for the introduction of the word " force " nor of the sense- suggested ideas on which it was originally based. 300. Nothing beyond a mere mention has been made above of virtual velocities, and of the so-called element ary machines. These belong to the subject of Applied Mechanics, separately treated below. 301. The references which have been made to various grand theories, such as action, impulse in general, &c., have been illustrated by simple cases only. For a detailed examination of these theories the reader is referred to Thomson and Tait s Natural Philosophy, chap. ii. To the same work he is referred for the general "theory of small oscillations," the "dissipative function," the "ignoration of coordinates," the treatment of "gyrostatic systems" and of " kinetic stability." All of these have been exhibited, though in mere particular instances, in the preceding pages. Treatises on Mechanics. 302. The following works on Mechanics are indispensable : 1. Newton s Principia (1st ed., 1687 ; latest ed. , Glasgow, 1871). Here, for the first time, the fundamental principles were systematized, extended (as we have seen) in a most vital particular, and applied, by the aid of a new mathematical method of immense power (based entirely on kinematical considerations), to many of the most important questions of cosmieal and terrestrial dynamics. Newton s system was first taught in the university of Edinburgh ; and, with brief intervals, his methods also have been habitually kept before the students there. From the time of Maclaurin to that of Forbes the value of the quasi-geometrical methods in giving a clear insight into the problems treated has rarely been overlooked. In Cambridge these methods were of later introduction, but they still deservedly figure as a necessary part of the reading of candi dates for "mathematical honours." It is to be feared, however, that in some other British universities the study of Newton s methods is not prosecuted to anything like the same extent. But the very reverse seems to be the case in America, where, probably to a considerable extent on this account, mathematical physics is advancing in a most remarkable manner. 2. Lagrange s Mecanique Analytique (1st ed., 1788). Though objections may fairly be taken to the fundamental method of this work, there can be no question as to the immense power and origin ality of its author. His "generalized coordinates," and the equations of motion of a system in terms of these, form one of the most im portant contributions to the science since the days of Newton. The method of Lagrange, though he was not aware of the fact, is really based upon the consideration of energy ; and when, in quite recent times, experiment had shown what are the grand laws of energy, Lagrange s magnificent mathematical methods and results were, ready for translation into the new language of science. 3. Hamilton s papers in the Philosophical Transactions for 1833 and 1834. Here the principle of varying action, and the characteristic function, were first applied to mechanics ; though they had been given, some years before, to the Royal Irish Academy, in their optical applications. Grand as have been the extensions of these new ideas made by Hamilton himself, and by many others, among whom Jacob! and Liouville may be especi ally mentioned, they have been mainly in a purely mathe matical direction. We wait for what cannot now be long delayed, the coming of the philosopher who is to tell us the true dynamical hearings of varying action and of the characteristic function. 4. If to these we add some of the works of Galileo, Huygens, Euler, Maclauriu, and D Alembert, we have the great landmarks in the history of the subject, as distinguished from its development. 5. The mere enumeration of the more important developments which the subject lias received, as distinguished from the absolutely new grand ideas and methods introduced, would require a long article. Brilliant examples of what may be done in this direction are furnished by Stokes s "Report on Recent Researches in Hydro dynamics" and by Cayh-y s " Reports on Theoretical Dynamics" (printed in the British Association Reports for 1846, and for 1857 and 1862). These should be consulted by every student who desires to trace the growth of the subject. They have been suc ceeded, in the same Reports (1880, 1881) by two excellent sum maries, by Hicks, of " Recent Progress in Hydrodynamics. " But Laplace s Mecanique Celeste, Poisson s Mecanique, Poinsot s Theorie Nouvelle de la Rotation, &c., more or less parts of the immediate outcome of the period when France intellectually dwarfed the rest of the world, are still of far more than mere historic value. For the English-reading student of modern times, the work of Thomson and Tait will be found suitable. The authors of this 749 work claim the position of "restorers," not of innovators; and they have (since 1863, when the first short sketch of their work was published) striven with success to re-establish in Britain Newton s grand yet simple foundations of the subject. Hut these- foundations, as stated above, are only temporarily the best. We have not, as yet, anything nearly so good. Other modern works of value are the Analytic Mechanics of the late Professor Peirce (Boston, 1855) and KirchhotFs VorUsunge.n iiber Mathematische Physik (Leipsic, 1876). Both are rather of t he- nature of collections of short treatises on special questions than organized wholes, but both will well repay careful reading. This, in the case of Peirce s work, is rendered extremely puzzling and laborious by the peculiar notations and modes of reference adopted by the author. It is particularly interesting to study the ways in which the fundamental principles are introduced in these works, and to compare them with the corresponding parts of the works of Newton and Lagrange. Lagrange, Peirce, and Kirchholf construct each a system as free from anything but analysis as possible. In fact Lagrange prefaces his work by the characteristic statement, " On ne trouvera point de Figures dans cet ouvrage. Les methodes quo j y expose ne demandent ni constructions, ni raisonnemens geometriques ou mechaniques, mais seulement des operations algebriques, assujetties a une marche reguliere et uniforme. Ceux. qui aiment FAnalyse verront avec plaisir la Mechanique en devenir une nouvelle branche." .... How far we have considered it ex pedient to differ from such an authority, a glance at the preceding pages will show. A part of the detailed work of several of the examples above given in Dynamics of a Particle has been taken from the elementary treatise (with that title) of Tait and Steele. The English reader who wishes to pursue* elementary Statics may profitably consult the treatise of Minchin. The higher parts are discussed in the work of Somoff, Thcorctischc Mcchanik (Leipsic, 1879). An excellent introduction to the use of Generalized Coordinates has been pub lished by Watson and Burbury (1879). On Lagrange s Generalized Equations the student should also read in Maxwell s Treatise on Electricity and Magnetism, part iv. chap. v. And Maxwell s brief treatise on Matter and Motion should be in the hands of every one commencing the subject. ANALYSIS OF THE PRECEDING ARTICLE. NEWTON S LAWS OF MOTION, with Comments, assumed as tlic basis of (he article, 1 13. KINEMATICS : Position, 14-19; Kinematics of Point. 20-70; of Plane Figure in its own Plane, 71-74 ; of Rigid Figure, 75-83 ; of Deformable Figure, 84-95. DYNAMICS OF A PARTICLE : General Considerations, 96-113 ; Further Comments on the First Two Laics of Motion, 114- 119; Friction, 120-121 ; Statics of a Particle, 122-128 ; Kinetics of a Particle ivith One Degree of Freedom (Meteorite, Hailstone, Pendulum, Cycloidal and Resisted Pendulum), 129 139 ; with Tico Degrees of Freedom (Planetary Mo tion, Kepler s Laics and their Consequences, Kinetic Stability], g 140-149; The Brachistochrone, 150-152 ; Kinetics of a Particle generally (Conical Pendulum, Blackburn s and Foil- cault s Pendulums, Varying Constraint, Disturbed Motion], 153-163; Third Law, Kinetics of Two or More Particles (Aticood s Machine, Chain-shot, Complex Pendulum], 164- 178 ; Kinetics of Free Particles generally, Virial, 179 ; Impact (Continuous Scries of Infinitely Small Impacts, Rocket], 180-190 ; Dynamics of a System of Particles generally (Equilibrium Neutral, Stable, and Unstable ; Lagrange s General Equation], 191-199; Action, 200-214; Gene ralized Coordinates, 215, 216. STATICS OF A RIGID SOLID : Reduction of Forces to Force and Couple? Minding s Theorem, Examples of Statical Problems, 217- 233. KINETICS OF A RIGID SOLID : Moment <>f Inertia, Binefs Theorem, Compound Pendulum, Ballistic Pendulum, Rolling and Slid ing of Sphere, Motion about Fixed Point, Poinsot s and Sylvester s Constructions, Quoit, Gyroscopic Pendulum, 234-257. STATICS OF A CHAIN : Common Catenary, Catenary of Uniform Strength, Kinetic Analogy, Chain Stretched on Surface, 258-264. KINETICS OF A CHAIN : Wan Propagation, Musical String, Cliain with One End Free, Impulsive Tension, Longitudinal Wave, 265-270. DYNAMICS OF ELASTIC SOLID : Flexure and Torsion of Wire, Bending of Plank, Oscillation of Flat Spring, Distortion of Cylinders and Spheres ly Internal and External Hydrostatic Pressure, 271-284. GENERAL CONSIDERATIONS ABOUT FORCE AND ENERGY: Newtona Idea of Force, Origin of the Conception, Stress, Objective Physical Realities, True Nature of Force, Rates in General, Potential Energy in its Nature Kinetic, Maxwell on Inertia, True Laic* of Motion, 285-299.
References to Authoritative Works, 301, 302.