A treatise on infinitesimal calculus, containing differential and integral calculus, calculus of variations, applications to algebra and geometry, and analytical mechanics, Volume 4 (Google eBook)

Front Cover
University Press, 1862 - Calculus of variations
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Contents

Composition of angular velocities whose rotationaxes meet
40
Particular forms when the rotationaxes are at right angles
41
Bohnenbergers machine
42
Composition of many angular velocities whose rotationaxes pass through one point
43
Geometrical explanation of Foucaults pendulum experiment
44
Composition of angular velocities whose rotationaxes are parallel
45
Couples of angular velocities
47
General case of the composition of angular velocities
48
Particular cases of the preceding Article
50
3537 The central axis
53
Determination of the linear velocity due to given angular ve locities
56
Motion of a body defined by two systems of reference
58
Relation between angular velocities and the differentials of directioncosines
60
Analytical proof that all motion consists of a translation and of a rotation
64
CHAPTER III
67
Explanation and statement of DAlemberts principle
69
Illustrations of its application to particular problems
73
47 The statement and proof given by DAlembert
77
The equations of motion of a material system the internal forces of which are in equilibrium
78
The equations of motion expressed in a single equation by means of the principle of virtual velocities
80
The number of equations necessary for the complete solution of a problem
83
Application to the motion of a flexible string
84
Independence of the motions of translation of centre of gra
86
Conservation of motion of centre of gravity of moments
92
Conservation of areas
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 equations deduced from first principles
130
The axial components of the resulting angular velocity
131
77 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
Simplification of the equations principal axes
139
Proof of the existence of a system of principal axes and the position of it at a given point
141
87 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
151
The central principal axes the only permanent axes
153
Foucaults gyroscope
154
General value of moment of inertia
156
9 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
109110 Particular forms of the equimomental surface
174
Distribution in space of principal axes 178
156
Conditions when a line is a principal axis
179
Examples of moments of inertia
185
CHAPTER V
199
Conditions when there is no pressure on the axis
206
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
CHAPTER VI
245
The equations of motion in their general and reduced forms
246
The instantaneous rotationaxis the instantaneous pole and the couple of impulsion
248
General and particular properties of the polhode
264
158159 Do of the herpolhode
266
The stability of the rotationaxis
269
Particular cases of the preceding theorems depending on particular initial circumstances
271
162163 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
Particular case when a b and the initial axis of rotation is the principal axis of unequal moment
288
170171 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
Simplification of the equations 301
139
Determination of small quantities
300
16 Transformation of certain terms of the equations
304
177 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
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 vis viva of the system is a maximum when the rotation axis is the spontaneous axis
327
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
332
The centres of greatest percussion
334
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
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
435
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
The results of the gyroscope 480
566
26 Moving work and resisting work useful work and lost
487
On mechanical units
493
Two modes of forming the equations of motion of an elastic
499
The properties of a vibrating string deduced from the pre
506
Another mode of expressing the integrals
513
The general equations of motion of a molecule of an elastic
519
THEORETICAL DYNAMICS
524
Case in which the equation 8 A 8 A is significant
530
Illustration taken from the transformation to polar coordi
536
Considerations on the integrals of such a system
544
Definition of canonical elements The initial values
550
If f g be any two integrals g is constant example
97
all the integral equations by means of a single function
317
Modification in the case in which principle of vis viva holds
324
canonical solution is reducible to quadratures
562
Interpretation of the three new elements
570
Conclusion
578

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Page 298 - Newton discovered, as a fundamental law of nature, that every particle attracts every other particle with a force which varies directly as the product of the masses and inversely as the square of the distance between them.
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Page 542 - This proves the first part of the theorem. To prove the second part : Take any two lines of the system, 34 and 56.
Page 89 - ... bottom of which are formed by planes perpendicular to its axis, contains elastic fluid, the weight of which may be neglected. If the vessel revolve uniformly about its axis, find the pressure at any point of the fluid mass. 6. The motion of rotation of a rigid system acted on by any forces, about its centre of gravity, is the same as if the centre of gravity were fixed, and the same forces acted. A heavy beam moves about a horizontal axis passing through one extremity ; apply the preceding principle...
Page 93 - Conservation of vis viva., the Principle of the Conservation of the Motion of the Centre of Gravity, and the like.
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Page 536 - ... x = r sin 6 cos <f>, y r sin 6 sin <f>, z = r cos 6.
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