Treatise on Natural Philosophy

Conservative System-Foundation of the Theory of Energy-
Physical axiom that "the Perpetual Motion is impossible"
introduced-Potential Energy of Conservative System
Inevitable loss of Energy of Visible Motions-Effect of Tidal
Friction-Ultimate tendency of the Solar System

Conservation of Energy

Kinetic Energy of a System-Moment of Inertia-Moment of
Momentum of a Rotating Rigid Body-Radius of Gyration
-Fly-wheel-Moment of Inertia about any Axis

Momental Ellipsoid-Equilibration of Centrifugal Forces-

Definition of Principal Axes of Inertia-Principal Axes-

Binet's Theorem-Central Ellipsoid-Kinetic Symmetry

round a Point; round an Axis

Energy in Abstract Dynamics

Equilibrium-Principle of Virtual Velocities-Neutral Equi-

librium-Stable Equilibrium-Unstable Equilibrium-

Test of the nature of Equilibrium

Deduction of the Equations of Motion of any System-Inde-

terminate Equation of Motion of any System-of Conser-

vative System-Equation of Energy-Constraint intro-

duced into the Indeterminate Equation-Determinate

Equations of Motion deduced-Gauss's Principle of Least

Constraint

SECTIONS

271-274

275-277

280, 281

294-306

Impact-Time-integral-Ballistic Pendulum-Direct Impact
of Spheres-Effect of Elasticity-Newton's Experiments—
Distribution of Energy after Impact-Newton's experi-
mental Law consistent with perfect Elasticity

Moment of an Impact about an Axis-Ballistic Pendulum---

Work done by Impact-Equations of Impulsive Motion 307-310

Theorem of Euler, extended by Lagrange-Liquid set in Motion

impulsively-Impulsive Motion referred to Generalized

Co-ordinates-Generalized Expression for Kinetic Energy

-Generalized Components of Force-of Impulse-Im-

pulsive Generation of Motion referred to Generalized

Co-ordinates-Momentums in terms of Velocities-Kinetic

Energy in terms of Momentums and Velocities - Velo-

cities in terms of Momentums--Reciprocal relation be-

tween Momentums and Velocities in two Motions-Ap-

plication of Generalized Co-ordinates to Theorems of

§ 311-Problems whose data involve Impulses and Velo-

cities-General Problem (compare § 312)-Kinetic Energy

a minimum in this case-Examples

Lagrange's Equations of Motion in terms of Generalized Co-

ordinates deduced direct by transformation from the

Equations of Motion in terms of Cartesian Co-ordinates

-Equation of Energy-Hamilton's Form-" Canonical

311-317

form" of Hamilton's general Equations of Motion of a

Conservative System-Examples of the use of Lagrange's

Generalized Equations of Motion-Gyroscopes and Gy-

rostats―Gyroscopic Pendulum-Ignoration of Co-ordi-

nates

Kinetics of a perfect fluid-Effect of a Rigid Plane on the

Motion of a Ball through a Liquid-Seeming Attraction

between two ships moving side by side in the same

direction-Quadrantal Pendulum defined-Motion of a

Solid of Revolution with its axis always in one plane through

a Liquid-Observed phenomena-Applications to Nautical

Dynamics and Gunnery Action Time Average of

Energy-Space Average of Momentums-Least Action—

Principle of Least Action applied to find Lagrange's

Generalized Equations of Motion--Why called "Station-

ary Action" by Hamilton Varying Action Action

expressed as a Function of Initial and Final Co-ordinates

and the Energy; its differential Coefficients equal re-
spectively to Initial and Final Momentums, and to the
time from beginning to end-Same Propositions for Ge-
neralized Co-ordinates-Hamilton's "Characteristic Equa-
tion" of Motion in Cartesian Co-ordinates-Hamilton's
Characteristic Equation of Motion in Generalized Co-or-
dinates-Proof that the Characteristic Equation defines
the Motion, for free particles-Same Proposition for a
Connected System, and Generalized Co-ordinates-Ha-
miltonian form of Lagrange's Generalized Equations de-
duced from Characteristic Equation .

Characteristic Function-Characteristic Equation of Motion-

Complete Integral of Characteristic Equation-General

Solution derived from complete Integral-Practical In-

terpretation of the complete Solution of the Characteristic

Equation-Properties of Surfaces of Equal Action-

Examples of Varying Action-Application to common

Optics or Kinetics of a Single Particle-Application to

System of free mutually influencing Particles--and to
Generalized System

Slightly disturbed Equilibrium-Simultaneous Transformation

of two Quadratic Functions to Sums of Squares-Gene-

ralized Orthogonal Transformation of Co-ordinates--Sim-

plified expressions for the Kinetic and Potential Energies

-Integrated Equations of Motion, expressing the fun-

damental modes of Vibration; or of falling away from

Configuration of Unstable Equilibrium-Infinitely small

Disturbance from Unstable Equilibrium-Potential and

Kinetic Energies expressed as Functions of Time--

Example of Fundamental Modes

SECTIONS

318, 319

320-330

331-336

337

General Theorem of Fundamental Modes of infinitely small

Motion about a Configuration of Equilibrium--Normal

Displacements from Equilibrium-Theorem of Kinetic

Energy-Of Potential Energy-Infinitesimal Motions in

neighbourhood of Configuration of Unstable Equilibrium

Case of Equality among Periods-Graphic Representation—

Dissipative Systems-Views of Stokes on Resistance to a

Solid moving through a Liquid - Friction of Solids-

Resistances varying as Velocities-Effect of Resistance

varying as Velocity in a simple Motion

Infinitely small Motion of a Dissipative System-Cycloidal

System defined-Positional and Motional Forces Differ-

ential Equations of Complex Cycloidal Motion-Their

Solution-Algebra of Linear Equations-Minors of a De-

terminant-Relations among the Minors of an Evanescent

Determinant-Case of Equal Roots-Case of Equal Roots

and Evanescent Minors-Routh's Theorem-Case of no

Motional Forces-Conservative Positional, and no Mo-

tional, Forces--Equation of Energy in Realized General

Artificial or Ideal Accumulative System-Criterion of Sta-
bility-Cycloidal System with Conservative Positional
Forces and Unrestricted Motional Forces-Dissipativity
defined-Lord Rayleigh's Theorem of Dissipativity-In-
tegral Equation of Energy-Real part of every Root of
Determinantal Equation proved negative when Potential
Energy is positive for all real Co-ordinates; positive for
some Roots when Potential Energy has negative values;
but always negative for some Roots-Non-oscillatory sub-
sidence to Stable Equilibrium, or falling away from Un-
stable-Oscillatory subsidence to Stable Equilibrium, or
falling away from Unstable-Falling away from wholly
Unstable Equilibrium is essentially non-oscillatory if
Motional Forces wholly viscous-Stability of Dissipative
System-Various origins of Gyroscopic Terms-Equation
of Energy-Gyrostatic Conservative System-simplifica-
tion of its Equations-Determinant of Gyrostatic Conser-
vative System-Square Roots of Skew Symmetric De-
terminants-Gyrostatic System with Two Freedoms-Gy-
rostatic Influence dominant-Gyrostatic Stability—Ordi-
nary Gyrostats-Gyrostats, on gimbals; on universal
flexure-joint in place of gimbals; on stilts; bifilarly slung in
four ways-Gyrostatic System with Three Freedoms-Re-
duced to a mere rotating System-Quadruply free Gyro-
static System without force-Excepted case of failing gy-
rostatic predominance-Quadruply free Cycloidal System,
gyrostatically dominated-Four Irrotational Stabilities
confirmed, four Irrotational Instabilities rendered stable,