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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.



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-
pocycloids, etc.

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
Body due to given Rotations

Most general Motion of a Rigid Body

Precessional Rotation-Model illustrating Precession of Equi-


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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
in Kinks

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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-
face of any are traced on it


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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”

Analogy between Lines and Surfaces as regards Curvature-
Horograph-Area of the Horograph




102, 103







126, 127


136, 137


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Analysis of a Strain into Distortion and Rotation
Pure Strain-Composition of Pure Strains
Displacement of a Curve-Tangential Displacement-Tan-
gential Displacement of a Closed Curve-Rotation of a
Rigid Closed Curve-Tangential Displacement in a Solid,
in terms of Components of Strain-Heterogeneous Strain
-Homogeneous Strain-Infinitely small Strain-Most
general Motion of Matter-Change of Position of a Rigid
Body-Non-rotational Strain-Displacement Function

"Equation of Continuity"-Integral Equation of Continuity-

Differential Equation of Continuity-"Steady Motion"


Generalized Co-ordinates- of a Point-of any system.
Generalized Components of Velocity-Examples

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180, 181







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