A Treatise on Infinitesimal Calculus: The dynamics of material systems. 1862

University Press, 1862

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PRELIMINARY GEOMETRICAL INVESTIGATIONS 1 The necessity of a geometrical investigation	1

Transformation of rectangular systems of reference in terms	2

nine directioncosines	3

Do in terms of Eulers three angles viz 0 p	4

Reduction of central surfaces of the second degree to the centre as origin	5

Do to principal axes as coordinate axes	6

Proof that the latter reduction is possible and determination of the position of the principal axes	7

Principal axes coincide with singular radii vectores	8

Do of the herpolhode	266

The stability of the rotationaxis	269

Particular cases of the preceding theorems depending on particular initial circumstances	271

Do depending on particular constitutions of the body	273

Discussion of the case when the oscillation of the rotation axis is small	277

The cone described by the rotationaxis in the body	278

Certain properties of the principal axes of the moving body	279

Rotation of a heavy body about a fixed point	281

Principal axes are perpendicular to their conjugate planes 9	9

28	18

Degenerate forms of surfaces of the second degree 11 Cones and reciprocal cones of the second degree 12 Cyclic planes of cones	20

Conjugate axes and planes in an ellipsoid 13 Cyclic planes of ellipsoids	21

The most general motion consists of a translation and of	22

On central radii vectores of an ellipsoid	26

The focal conics of an ellipsoid 18 Confocal surfaces of the second degree	28

CHAPTER II	32

PRICE VOL IV	33

Composition of many angular velocities whose rotationaxes	43

Motion of a body defined by two systems of reference	57

Analytical proof that all motion consists of a translation	64

the equations of motion of a	78

Independence of the motions of translation of centre of gra	86

Conservation of motion of centre of gravity of moments	92

Various forms of material systems and their definitions	94

The invariable plane	98

Conservation of vis viva critical value of vis viva when	105

Carnots theorem	114

CHAPTER IV	124

Transformation of equations instantaneous forces	127

Moment of inertia and radius of gyration	128

The number of equations necessary for the complete solution	129

The equations deduced from first principles	130

The axial components of the resulting angular velocity	131

Transformation of equations finite forces	132

The resulting angular velocityincrement	133

Axial components of velocityincrement	134

Analysis of the equations	135

Centrifugal forces	137

Resulting equations of motion	138

Application to the motion of a flexible string	139

Principal axes and their properties 86 Proof of the existence of a system of principal axes and the position of it at a given point	141

Interpretation of the results by means of an ellipsoid	143

Particular forms of the ellipsoid of principal axes	145

Principal axes determined in a particular problem	146

One principal axis being given the determination of the two others	147

Examples in illustration	148

Reduced forms of equations of motion	150

Permanent axes	153

Foucaults gyroscope	154

General value of moment of inertia	156

Example in illustration	157

Singular values of principal moments	160

The equimomental cone	161

Moments of inertia relative to parallel axes	162

Moments of inertia in relation to the central ellipsoid	163

The position of principal axes at any point	166

The central ellipsoid of gyration	167

The symmetry of a body	169

The cone reciprocal to that of principal axes is equimomental	171

Professor Maccullaghs construction by Apsidals	173

Particular forms of the equimomental surface	174

Distribution in space of principal axes	178

Examples of moments of inertia	185

Moments of inertia of shells derived from those of solid	197

THE ROTATION OF A BODY ABOUT A FIXED POINT SECTION 1 Rotation of a body about a fixed point under the action of instantaneous forces	199

Determination of the pressure on the axis	204

The axis of percussion	212

Examples of the simple isochronous pendulum	220

Experimental determination of the radius of gyration	226

Motion of machines with fixed axes	233

Two systems of reference	245

The equations of motion in their general and reduced forms	246

The instantaneous rotationaxis the instantaneous pole and the couple of impulsion	247

Examples in illustration	250

The pressure at the fixed point	252

Rotation of a rigid body about a fixed point under the action of finite forces 148 The general equations of motion Eulers equations	253

The equations and their results when no forces act	255

Explanations of these results the invariable axis	256

Position determined by the three angles 0 4 4	258

The component of the instantaneous angular velocity along the invariable axis is constant	260

Poinsots interpretation of the preceding result	261

General and particular properties of the polhode	264

Particular case when A B and the initial axis of rotation is the principal axis of unequal moment	288

Do when the axis of unequal moment is inclined to the vertical at a constant angle	291

Bohnenbergers and Fessels machines	295

Precession and nutation of the earth	297

Determination of small quantities	300

Transformation of certain terms of the equations	304

General integral of the equations	306

Determination of the lunisolar precession and nutation	311

Poinsots determination of the preceding results from gene ral reasoning	315

If f g be any two integrals f g is constant example	316

The pressure on the fixed point	318

CHAPTER VII	320

The equations of motion	322

The components of velocity resulting from a given force	323

The locus of points which move with the same velocity	324

The spontaneous axis	326

The motion of a body due to a blow parallel to a central principal axis and in a central principal plane	329

Properties of centre of percussion and spontaneous centre of rotation	331

The centres of greatest percussion	333

The body struck may be equivalently replaced by two mole cules of given masses at the ends of an inflexible bar	335

The position of the point on which the moving body	338

The centres of greatest reflexion and greatest conversion	339

Motion of a body due to a blow parallel to a central princi pal axis	342

Theorems relating to the spontaneous axis and spontaneous centre corresponding to a given centre of impulsion	344

Other incidents of the motion	346

Centres of greatest percussion	349

Case in which a couple of impulsion initially acts	350

199200 Case when the body strikes against a moveable mass	352

Points of greatest reflexion and greatest conversion	355

Points of perfect reflexion and perfect conversion	356

The initial motion of a billiard ball	363

The general case of rocking or titubation	379

Examples of small oscillations	385

Small oscillations of a body of which one point is fixed	391

The motion of a top on a smooth horizontal plane	402

CHAPTER VIII	413

The equations derived from fictitious forces	420

Motion of a heavy particle in a rotating tube	429

Adaptation of the equations with the omission of small	436

The investigation carried to a higher approximation	442

Relative motion of a particle on a smooth inclined plane	453

All combined into one equation by means of the principle	459

Relations between absolute and relative angular velocities	465

Particular cases of the preceding	474

CHAPTER IX	481

Definition of a machine its working point its work and la bouring force	483

Relation between work and vis viva	485

Maximum and minimum values of vis viva	489

Uniformity of motion secured by flywheels	491

On mechanical units	493

CHAPTER X	496

A rigid body in contrast with an elastic body	498

Two modes of forming the equations of motion of an elastic body	499

The equations of motion of a fine flexible string	500

The general motion of an elastic string	501

The integrals of the equations of motion expressed in terms of arbitrary functions	504

The properties of a vibrating string deduced from the pre ceding functions	506

The oscillations and periodic times of the string	508

The nodes and ventral segments	509

The relation between the periodic times of the transverse and longitudinal vibrations	510

The preceding results in reference to the theory of music	511

Another mode of expressing the integrals	513

The longitudinal vibrations of the molecules of a fine elastic rod	514

The motion of the molecules of a thin elastic lamina	517

The general equations of motion of a molecule of an elastic body	519

Particular case when the body is a thin elastic membrane	520

Particular case when the body is a fine elastic string	522

THEORETICAL DYNAMICS	524

Proof that 8 u 8 u	531

Second example transformation from fixed to moving rect	537

Definition of canonical elements The initial values	550

The results of the gyroscope 480	566

Simplification in the case of canonical elements	574

Otras ediciones - Ver todo

A Treatise on Infinitesimal Calculus: The dynamics of material systems. 1862
Bartholomew Price
Vista completa - 1889

A Treatise on Infinitesimal Calculus: The dynamics of material systems. 1862
Bartholomew Price
Vista completa - 1862

Términos y frases comunes

a₁ angle angular velocity axial components axis passing b₁ blow body rotates central ellipsoid central principal centre of gravity cone confocal coordinate axes couple D'Alembert's principle determined direction-cosines displacement distance dt dt dt dt2 dt2 earth ellipse ellipsoid of gyration equal equations of motion expressed fixed point focal conic given impressed momentum-increments infinitesimal invariable plane isochronous Let the rotation-axis let us suppose line of action mass moment of inertia momental ellipsoid moments of inertia momentum Mp² origin oscillation P₁ parallel pendulum perpendicular plate polhode position principal axes principal moments principal plane quantities radii radius of gyration radius vector relative second degree shews sphere string surface system of particles theorem tion values velocity-increment vertex vertical vis viva viva w₁ w₂

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Información bibliográfica

Título	A Treatise on Infinitesimal Calculus: The dynamics of material systems. 1862 Volumen 4 de A Treatise on Infinitesimal Calculus: Containing Differential and Integral Calculus, Calculus of Variations, Applications to Algebra and Geometry, and Analytical Mechanics, Bartholomew Price
Autor	Bartholomew Price
Edición	2
Editor	University Press, 1862
Procedencia del original	la Universidad de California
Digitalizado	5 Jun 2007

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