The Mathematical Principles of Mechanical Philosophy and Their Application to Elementary Mechanics and Architecture: But Chiefly to the Theory of Universal Gravitation

Front Cover
J. & J.J. Deighton, 1842 - 620 pages
 

Contents

31
71
35
77
ARTICLE PAGE
78
Pullies Whites Pully 113 Inclined Plane 114 Wedge
86
CHAPTER VI
91
Explanation of the action of buttressesRoofs of Trinity College
92
Equilibrium of an arch when voussoirs are smooth
98
Conditions of equilibrium of an arch that it may not break
104
CHAPTER III
111
Common Catenary Property of its centre of gravity Tension
112
141
116
Conditions of equilibrium
122
Principle of Virtual Velocities
132
CHAPTER VIII
139
The laws for which a shell attracts as if condensed into its centre
143
Attraction of a homogeneous oblate spheroid of any ellipticity on
149
Ditto for an internal particle
150
Attraction of a body on a very distant particle the same nearly
155
Laplaces Coefficients
170
ARTICLE PAGE
177
A function of μ 1µ³ cos w 12 sin w can be expanded
181
Dynamical measure of force
183
Measure of impulsive accelerating force
190
The necessity of a third law to establish the relation between
196
Means of obtaining equations to calculate motion
205
Rectilinear motion under the action of gravity and central forces
211
Curvilinear motion under the action of gravity
217
Properties of central orbits
223
ARTICLE PAGE
225
The law of force found when the orbit is known Force in
231
CHAPTER III
241
Method of determining the elements of an orbit from observation
247
Time in a parabolic orbit about the focus
254
Brief history of lunar inequalities
260
ARTICLE PAGE 299 Effect on the periodic time of the Moon
265
Effect on the velocity in the circular orbit
266
Effect on the form of the circular orbit
267
The ratio of the axes of the oval orbit
268
The velocity in the oval orbit Moons Variation
269
Effect on the inclination and line of nodes
271
Calculation of the motion of the nodes
273
Calculation of the inclination of the Moons orbit
277
CHAPTER V
278
The equations of motion of a body attracted by other bodies
279
PLANETARY THEORY
282
Equations for calculating the distance of the Moon and the incli nation of the lunar orbit to the ecliptic in terms of her longitude
283
Comparison of small quantities
286
Expansions of P T S to the third order
287
Integration of the equations first approximation
289
Reason why some terms must be calculated to the third order for the second approximation
290
Calculations for a second approximation
291
introduction of the constants c and g
294
Integration of the equations second approximation
297
Distance of the Moon from the Earth
298
Geometrical interpretation of the terms in the formulæ for the dist ance and longitude of the Moon Progression of the line of Apsides The Variation T...
303
Integration of the equations for a disturbed planet
318
Transformation of the differential coefficients of R
321
Variations of the elements
324
Method of expanding R
329
The form of the terms and the order of magnitude of the co efficients in this expansion The constant part of R
331
Effect of the terms of R after the first periodical
338
Jupiter and Saturn the Earth and Venus
340
Difference between periodic and secular variations
341
Equations for calculating the secular variations
342
and of the eccentricities and inclinations The fact that the planets re volve about the Sun in the same direction ensures the Stability of the System
345
The masses and elements of the heavenly bodies
351
CHAPTER VII
357
oscillations are isochronous
359
Motion on a circular arc
361
measuring of heights and depths
363
Oscillation of a pendulum in a cycloid
365
Table of the lengths of the seconds pendulum in various places
366
ARTICLE PAGE
375
CHAPTER VIII
381
Formula for the transformation of coordinates
387
Properties of the principal moments of inertia
395
MOTION ABOUT A FIXED AXIS The angular accelerating force
401
Equations for calculating the angular velocities about the principal
413
Stability of the Earths rotation
423
The Precession of the Equinoxes and the Nutation of the Earths
429
The inclination of the Earths axis to the lunar orbit is nearly
435
Plane of Principal Moments and Invariable Plane The action
442
ARTICLE PAGE
445
Conditions of stable and unstable equilibrium
450
Sir W Hamiltons Principle of Varying Action
452
Coexistence of Small Vibra
460
CHAPTER XIII
469
Impact of spheres
475
Motion of centre of gravity Conservation of Motion of Centre
481
Lagranges proof of Virtual Velocities
506
HYDROSTATICS
509
Internal arrangement of a mass of fluid in equilibrium Surfaces
516
Heterogeneous figure of the Earth The equation for calculating
525
ARTICLE PAGE
530
Calculation of the ellipticity
537
Additional theorems
543
CHAPTER IV
549
HYDRODYNAMICS
555
Equations for calculating the motion of an elastic fluid the
561
The difficulty of the subject of the Tides requires some hypothesis
563
Guldinus Properties
568
Transformation of the equations to polar coordinates
570
CHAPTER V
577
CHAPTER III
579
APPENDIX
599
174
605
the figure is
606
Problems on Terrestrial Magnetism
615
Equilibrium of the lever pressure on the fulcrum

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Page 505 - A uniform ladder, 10 feet long, rests with one end against a smooth vertical wall and the other on the ground, the coefficient of friction between the ladder and the ground being J.
Page x - D'Alembert, was the Precession of the equinoxes and the Nutation of the earth's axis, according to the theory of gravitation.
Page 238 - Gravitation is, that every particle of matter attracts every other particle with a force which varies directly as the mass of the attracting particle, and inversely as the square of the distance.
Page 4 - ... to be statical. But if the force be estimated by the magnitude of the motion generated in a body which it causes to move, the estimate is dynamical. 10. Weight is the name given to the pressure which the attraction of the Earth causes a body to exert on another with which it is in contact. Since the gravitation of bodies downwards is unceasing, weight becomes a very useful means of estimating all statical forces. Thus the force of a constrained spring, may be measured by the weight which will...
Page 85 - P act vertically, e - ^ir - a, P - W. 114. The fifth Mechanical Power is the Wedge. This is a triangular prism, and is used to separate obstacles by introducing its edge between them and then thrusting the wedge forward. This is effected by the blow of a hammer or other such means, which produces a violent pressure for a short time, sufficient to overcome the greatest forces.
Page 71 - A Lever is an inflexible rod moveable only about a fixed axis, which is called the fulcrum. The portions of the lever into which the fulcrum divides it are called the arms of the lever : when the arms are in the same straight line, it is called a straight lever, and in other cases a bent lever.
Page 523 - Let us express a in terms of the ratio of the centrifugal force at the equator to the equatorial gravity. Call this ratio m, which is small in the case of the earth, being of the same order as g.
Page 231 - Put 9 = 0 and r = R in the value of - : r By referring to Art. 234, we see that the velocity of a body falling from an infinite distance to a distance R from a centre of force — is equal to \/ —. Hence the orbit described about this centre of force will be an ellipse, parabola, or hyperbola according as the velocity is less than, equal to, or greater than that from infinity. 253. We might make use of the equation to discover the law of force when the orbit is given. Thus if the orbit be a conic...
Page 192 - Mr. Hodgkinson proceeds to describe the methods by which his experiments were made, and derives from them the following nonelusions : 1. All rigid bodies are possessed of some degree of elasticity, and among bodies of the same nature the hardest are generally the most elastic.
Page 385 - A body moves in a groove under the action of two centres of force each varying inversely as the distance, and of equal intensity at the same distance; the body is projected from the mid-point between the centres: prove that if the velocity be uniform the form of the groove is a lemniscate. PROB. 32. A body attracted to two centres of force varying inversely as the square of the distance moves in a hyperbolic groove, of which the foci are the centres of force : required to find the pressure on the...

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