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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 chapter.
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.
P. G. TAIT.
Displacements of a Plane Figure in its Plane--Composition
Motion of a Rigid Body about a Fixed Point-Euler's Theorem
Most general Motion of a Rigid Body
Precessional Rotation-Model illustrating Precession of Equi-
Free rotation of a Body kinetically symmetrical about an axis
Surface rolling on Surface; both traces given
Surface rolling on Surface without spinning
Curvature of Surface-Synclastic and Anticlastic Surfaces-
Integral Curvature-Curvatura integra-Horograph-Change
of direction round the boundary in the surface, together
Analogy between Lines and Surfaces as regards Curvature-
Flexible and Inextensible Surface-Flexure of inextensible
Analysis of a Strain into Distortion and Rotation
Generalized Co-ordinates- of a Point-of any system.