one of the most important of all analytical results as regards usefulness in physical science. In the Appendices to that chapter we have introduced an extension of Green's Theorem, and a treatise on the remarkable functions known as Laplace's Coefficients. There can be but one opinion as to the beauty and utility of this analysis of Laplace; but the manner in which it has been hitherto presented has seemed repulsive to the ablest mathematicians, and difficult to ordinary mathematical students. In the simplified and symmetrical form in which we give it, it will be found quite within the reach of readers moderately familiar with modern mathematical methods. In the second chapter we give Newton's Laws of Motion in his own words, and with some of his own comments—every attempt that has yet been made to supersede them having ended in utter failure. Perhaps nothing so simple, and at the same time so comprehensive, has ever been given as the foundation of a system in any of the sciences. The dynamical use of the Generalized Coordinates of LAGRANGE, and the Varying Action of HAMILTON, with kindred matter, complete the cbapter. The third chapter, “ Experience,” treats briefly of Observation and Experiment as the basis of Natural Philosophy. The fourth chapter deals with the fundamental Units, and the chief Instruments used for the measurement of Time, Space, and Force. Thus closes the First Division of the work, which is strictly preliminary, and to which we have limited the present issue. This new edition has been thoroughly revised, and very considerably extended. The more important additions are to be found in the Appendices to the first chapter, especially that devoted to Laplace's Coefficients; also at the end of the second chapter, where a very full investigation of the "cycloidal motion" of systems is now given; and in Appendix B', which describes a number of continuous calculating machines invented and constructed since the publication of our first edition. A great improvement has been made in the treatment of Lagrange's Generalized Equations of Motion. We believe that the mathematical reader will especially profit by a perusal of the large type portion of this volume; as he will thus be forced to think out for himself what he has been too often accustomed to reach by a mere mechanical application of analysis. Nothing can be more fatal to progress than a too confident reliance on mathematical symbols; for the student is only too apt to take the easier course, and consider the formula and not the fact as the physical reality. In issuing this new edition, of a work which has been for several years out of print, we recognise with legitimate satisfaction the very great improvement which has recently taken place in the more elementary works on Dynamics published in this country, and which we cannot but attribute, in great part, to our having effectually recalled to its deserved position Newton's system of elementary definitions, and Laws of Motion. We are much indebted to Mr BURNSIDE and Prof. CHRYSTAL for the pains they have taken in reading proofs and verifying formulas; and we confidently hope that few erratums of serious consequence will now be found in the work, W. THOMSON. P. G. TAIT. CONTENTS. Curvature and Tortuosity of a Tortuous Curve Integral Curvature of a Curve (compare & 136) Flexible Line--Cord in Mechanism Resolution and Composition of Accelerations Determination of the Motion from given Velocity or Ac- Acceleration directed to a Fixed Centre Angular Velocity and Acceleration Composition of Simple Harmonic Motions in one Line Composition of two Uniform Circular Motions Fourier's Theorem 63--74 75-77 SECTIONS Displacements of a Plane Figure in its Plane—Composition of Rotations about Parallel Axes-Composition of Rota- tions and Translations in One Plane-Superposition of Small Motions—Rolling of Curve on Curve-Cycloids and Trochoids—Properties of the Cycloid—Epicycloid, Hy- Motion of a Rigid Body about a Fixed Point-Euler's Theorem -Rodrigues' Co-ordinates—Composition of Rotations- Composition of Angular Velocities—Composition of suc- cessive Finite Rotations—Rolling Cones-Position of the Most general Motion of a Rigid Body Precessional Rotation Model illustrating Precession of Equi- Free rotation of a Body kinetically symmetrical about an axis Communication of Angular Velocity equally between Inclined Axes—Hooke's Joint Universal Flexure Joint-Elastic Universal Flexure Joint — Moving Body attached to a Fixed Object by a Universal Flexure Joint-Two Degrees of Freedom to move enjoyed by a Body thus suspended. General Motion of one Rigid Body touching another-Curve rolling on Curve-Plane Curves not in same Plane- Curve rolling on Curve; two degrees of freedom-Curve rolling on Surface; three degrees of freedom - Trace prescribed and no Spinning permitted; two degrees of freedom -- Angular Velocity of direct Rolling - Angular Velocity round Tangent – Surface on Surface - Both traces prescribed ; one degree of freedom Twist Estimation of Integral Twist in a Plane Curve; in a Curve consisting of plane portions in different Planes; in a continuously Tortuous Curve—Dynamics of Twist Surface rolling on Surface; both traces given Surface rolling on Surface without spinning Examples of Tortuosity and Twist Curvature of Surface-Synclastic and Anticlastic Surfaces- Meunier's Theorem - Euler's Theorem - Definition of Line of Curvature-Shortest Line between two points on a Surface - Spherical Excess Area of Spherical Polygon-Reciprocal Polars on a Sphere-Integral change of direction in a Surface-Change of direction in a Sur- Integral Curvature-Curvatura integra-Horograph--Change of direction round the boundary in the surface, together with area of horograph, equals four right angles: or "In- tegral Curvature” equals “Curvatura integra" 139-153 Flexible and Inextensible Surface-Flexure of inextensible Developable-Edge of Regression-Practical Construction vature-Geodetic Triangles on such a Surface Strain-Definition of Homogeneous Strain Properties of Ho- mogeneous Strain-Strain Ellipsoid—Change of Volume -Axes of a Strain-Elongation and Change of Direction of any Line of the Body-Change of Plane in the Body- Conical Surface of equal elongation—Two Planes of no distortion, being the Circular Sections of the Strain Ellip- soid-Distortion in Parallel Planes without Change of Volume-Initial and altered Position of Lines of no Elongation—Simple Shear-Axes of a Shear—Measure Analysis of a Strain into Distortion and Rotation 182 183-185 1864-190 Pure Strain-Composition of Pure Strains gential Displacement of a Closed Curve-Rotation of a Body-Non-rotational Strain-Displacement Function Differential Equation of Continuity—“Steady Motion" defined. Freedom and Constraint-Of a Point-of a Rigid Body-Geo- metrical Clamp--Geometrical Slide-Examples of Geo- metrical Slide-Examples of Geometrical Clamps and Slides_One Degree of Constraint of the most general character—Mechanical Illustration-One Degree of Con- straint expressed analytically Generalized Co-ordinates - of a Point--of any system Generalized Components of Velocity-Examples 191-194 195–201 202--201 APPENDIX Ag.-Expression in Generalized Co-ordinates for Poisson's Extension of Laplace's Equation. APPENDIX A. --Extension of Green's Theorem. APPENDIX B.-Spherical Harmonic Analysis. |