A treatise on infinitesimal calculus, Volume 3

Couverture
The University Press, 1856
 

Pages sélectionnées

Table des matières

Problems in illustration
60
Equilibrium of many bodies under the action of given forces
66
Examples of the preceding
78
On Friction
87
Reduction of all the forces to two forces of translation
94
The equations of the central axis
100
Relation between the central axis and the principal rotation
106
Such straight lines have reciprocal properties and are called
113
tion whose lines of action are perpendicular to each other
120
The directioncosines of the equilibriumaxis
128
The principal axis of rotation
135
Stability and Instability of Equilibrium 97 Explanation of stability neutrality continuity instability of equilibrium
139
Case of two forces
140
Case of forces acting in one plane
142
Case of many forces acting in space
143
Geometrical interpretation of the conditions
146
Some remarkable properties of the conditions
147
The principle of virtual velocities is deduced
149
The theorem of Gauss is also deduced
150
its variations
153
Weight density specific density specific gravity
154
Centre of gravity Its coordinates
157
Another definition of centre of gravity
159
The centre of gravity of material lines or wires 110 Investigation of the coordinates in this particular case
161
Examples in illustration
162
Application to curved wires in space
164
The curve which a heavy and flexible wire takes when its centre of gravity is in the lowest position
165
The first theorem of Pappus
167
Centre of gravity of thin plates and shells 115 Investigation of the coordinates of the centre of gravity in reference to rectangular coordinates in one ...
169
The same in reference to polar coordinates
172
Centre of gravity of a thin shell of revolution
173
Centre of gravity of a thin curved shell
175
The second theorem of Pappus
176
Centre of Gravity of heavy bodies 120 Investigation of the coordinates of the centre of gravity of a solid body bounded by a surface of revolution
178
The same for a solid body bounded by any curved surface
181
The same in reference to polar coordinates
182
Various examples
183
A position of equilibrium is stable neutral or unstable
185
THE ACTION OF FORCES ON BODIES OF VARIABLE form
194
The catenarian curve on a surface
200
The equation of the catenary
206
The stringcurve under the action of central forces
212
Hookes law
219
The extensible catenary
223
The action of molecular forces
225
Elastic forces
226
The equations of equilibrium of an elementary parallelepi pedon under the action of forces of elasticity
228
Illustrative examples 346
229
The same at the bounding surface of the body
230
The same when extended to the whole body
231
Calculation of the elastic forces which act on a massele ment of a body
232
General form of such elastic forces
234
Application to a homogeneous body of uniform elastic force
235
Simplification of the general equations by the considerations of the preceding Articles
238
The determination of certain constants
240
The bending of an elastic lamina
241
The equation to the curves of the fibres
244
Two particular cases
245
The vertical strength of a spring or a beam
247
The deflexion of a beam bent by its own weight
249
ON ATTRACTIONS 169 Introductory and explanatory
251
The mathematical expression for the attraction of two par ticles
252
The attraction of a straight rod or wire on an external particle
253
A remarkable geometrical construction of the result
254
The attraction of a bent rod or bar on an external particle
256
The attraction of a cylindrical tube on a particle in its axis
258
The attraction of a circular plate on a particle in the per pendicular through its centre
259
The attraction of a solid of revolution on a particle in its axis
260
The form of the solid of revolution of greatest attraction
263
The attraction of a homogeneous spherical shell on an ex ternal particle
264
The attraction of a sphere 1 homogeneous 2 hetero geneous
266
What are the laws for which the attraction of a shell on an external particle is the same as if the shell were condensed into its centre?
267
ticle within it
269
What are the laws for which the attraction of a spherical shell on a particle within it is zero?
271
If the attraction varies as the distance directly the resultant attraction is the same as if the body were collected into its centre of gravity
272
The attraction of a homogeneous ellipsoid
273
The attraction of an ellipsoidal shell
276
The attraction of spheroids
277
Ivorys theorem
279
Maclaurins theorem on attractions
282
The potential of attraction
283
The equilibriumsurface
284
Applications of the theory of the potential to two problems
286
The attraction of an ellipsoidal shell on an external particle by the method of M Chasles
291
Attraction of an ellipsoid on an external particle
297
The general properties of the potential
298
Investigation of the potential in certain cases
299
A projectile in a medium the resistance of which varies
300
PART II
302
Motion the fundamental idea matter its mobility and divisibility
305
Velocity constant and varying
306
The inertia of matter
308
The force varies inversely as the square of the distance
356
Rectilinear motion of particles in resisting media
362
Motion of a particle in a medium of which the density varies
368
Experimental illustrations of the law
374
The same problem when the resistance varies as the square
379
impulsive
380
Normal and tangential resolution of velocities
381
Centripetal and centrifugal forces
384
Radial and transversal resolution
385
Investigation of centrifugal force from first principles
387
The diminution of gravity on the earths surface as due to centrifugal force
388
The equations of relative motion
390
THE PRECEDING PRINCIPLES APPLIED TO THE MOTION OF PARTICLES IN SPACE SECTION 1 Oblique impact and collision of particles an...
392
Oblique impact on a smooth plane
393
Illustrative examples
395
Oblique impact of two particles or balls m and m
396
Oblique impact of elastic balls
397
The line of motion of the centre of gravity is not changed by the collision
398
Illustrative examples
399
The oblique effects of a resisting medium
400
The resistance of a fluid on a surface of revolution
401
The form of the surface when the resistance is a minimum
403
General investigations of the motion of a heavy particle on a smooth inclined plane
404
The synchronism of a circle in a vertical or in an inclined plane
407
The determination of planes of quickest and slowest descent
408
Illustrative examples of the motion of a particle on an in clined plane
409
Motion of two particles connected by a string passing over a pulley at the common vertex of two inclined planes
411
Illustrative examples
413
The process is most aptly indicated by examples Examples
414
Determination of the paths when the laws of force are given
421
Examples in illustration
427
An example of radial and transversal resolution
434
The centre of gravity of two such particles either remains
440
Determination of the law of resistance so that a given curve
446
If the resistance is small the path is parabolic
451
The sectorial areas vary as the times in which they are
457
Central conics the centre of force is in the centre
465
Other examples
472
The orbits when the force varies inversely as the square
478
The parabola
485
power of the distance
490
Some general properties of central orbits
491
Other cases which admit of integration in finite terms
493
Problems in illustration
494
The elements of Physical Astronomy 330 Observation is required ere the results of the science of mo tion can be applied to physical astronomy
500
The laws of Kepler their mathematical interpretation
503
The plane of the ecliptic The equinoxes the seasons
506
nutation the first point of Aries
509
generally seven
510
The radius vector and longitude in terms of the time
511
The corresponding expansions by means of Lagranges theorem
513
The equation of the centre
515
The time in the parabolic orbit
516
An approximate determination of the masses of the planets
518
The general differential equations of a disturbed body
520
The first approximate solution of the equation in latitude
522
Two cases where the equations admit of integration
524
the three bodies are always in a straight line
525
the system is shewn to be unstable
526
the three bodies are always in the angles of a revolving equilateral triangle
531
The theorem of M Bonnet
532
THE CONSTRAINED MOTION OF PARTICLES SECTION 1 A particle constrained to move on a given curved line 348 A normal pressure always ...
534
The equations when the motion is in one plane
537
Some examples in application of the principles
539
The equations of motion in the case of a heavy particle
542
The motion of a heavy particle on a cycloid
543
The cycloidal pendulum
545
Another solution of motion on a cycloid
546
The expression for the time of an oscillation
549
Another solution of the motion of a heavy particle in a
551
Examples illustrative of constrained motion
553
The general problem of tautochronism
555
Examples illustrative of tautochronism
557
Synchronous curves
559
Brachistochronism the general equation of brachisto chronous curves
560
Some particular cases of brachistochronous curves
562
Motion of particles in moving tubes
564
and general properties
567
The motion of a heavy particle in a sphere
569
The motion of a pendulum in a spherical surface
571
The motion of a particle on a surface of revolution
574
The circumstances under which a parallel of latitude is de scribed
575
The brachistochron on a given surface
576
Lines of easy motion on a surface
577
The relations between a brachistochron a geodesic line and a line of easy motion
581
Solution of some particular problems the cycloidal pendu lum in a resisting medium
582
The circular pendulum in a resisting medium
585
The tautochronous curve in a resisting medium
588
The nature of the forces when the principle is applicable
594
The problem of the path of the projectile in vacuo is solved
600
Also to the problem of the path of a projectile
606
Convention as to signs c
612

Autres éditions - Tout afficher

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Expressions et termes fréquents

accelerating force angle attracted particle axis balls body bounding surface catenary centre of gravity centrifugal force components confocal constant coordinate axes curve d2x dt2 determine direction displacement distance dt2 dt2 dx dy dz dy dx earth elastic force element ellipsoid equal equilibrium expressed velocity external particle finite forces acting given point Hence homogeneous horizontal impact impressed infinitesimal integrating investigate law of attraction length let us suppose line of action m₁ mass material particle matter molecules moment-centre momentum momentum-increment normal origin parallel perpendicular phænomena plane plate position pressures quantity r₁ radius resolved rest resultant shell space described sphere square straight line string theorem thickness and density tion unit varies directly varies inversely velo velocity vertical vertical strength vis viva weight αξ

Fréquemment cités

Page 5 - Admission to its sanctuary, and to the privileges and feelings of a- votary, is only to be gained by one means — sound and sufficient knowledge of mathematics, the great instrument of all exact inquiry, without which no man can ever make such advances in this or any other of the higher departments of science as can entitle him to form an independent opinion on any subject of discussion within their range.‎
Cité dans 84 livres de 1833 à 2007
Page 497 - When we contemplate the constituents of the planetary system from the point of view which this relation affords us, it is no longer mere analogy which strikes us — no longer a general resemblance among them, as individuals independent of each other, and circulating about the sun, each according to its own peculiar nature, and connected with it by its own peculiar tie. The resemblance is now perceived to be a true family likeness ; they are bound up in one chain — interwoven in one web of mutual...‎
Cité dans 59 livres de 1833 à 2006
Page 61 - ... 7, 6, and 9 pounds respectively at the points A, B, D, E, F; AB = 3 feet, BD = 6 feet, DE = 5 feet, EF = 4 feet. Find the magnitude of the resultant, and the distance of its point of application, G, from A. Ans. R — 42 pounds. AG = 8| feet.‎
Cité dans 7 livres de 1856 à 1995
Page 518 - ... the squares of the periodic times are as the cubes of the distances from the common centre, the centripetal forces will be inversely as the squares of the distances.‎
Cité dans 97 livres de 1715 à 2008
Page 415 - V (5) <7 which is the equation of a parabola with its axis vertical and the vertex the highest point of the curve. The distance, OB, between the point of projection and the point where the projectile strikes the horizontal plane is called the Range on the horizontal plane, and is the value of x when y = 0. Putting y = 0 in (3) of Art.‎
Cité dans 13 livres de 1856 à 2007
Page 497 - Of all the laws to which induction from pure observation has ever conducted man, this third law (as it is called) of Kepler may justly be regarded as the most remarkable, and the most pregnant with important consequences.‎
Cité dans 20 livres de 1833 à 2002
Page 243 - Every particle of matter in the universe attracts every other particle with a force that varies directly as the product of the masses of the particles and inversely as the square of the distance between them.‎
Cité dans 37 livres de 1856 à 2006
Page 499 - In this case, it is obvious that the plane of the circle of illumination would be perpendicular to a line drawn from the centre of the sun to the centre of the earth...‎
Cité dans 74 livres de 1728 à 2006
Page 429 - ... force between bodies as varying directly as their masses and inversely as the square of the distance between them; and would predict the elliptical planetary orbits that are in fact found in the solar system.‎
Cité dans 25 livres de 1809 à 2007
Page 254 - ... bodies is proportional to the product of their masses divided by the square of the distance between them.‎
Cité dans 105 livres de 1855 à 2007

Informations bibliographiques

TitreA treatise on infinitesimal calculus, Volume 3
A Treatise on Infinitesimal Calculus: Containing Differential and Integral Calculus, Calculus of Variations, Applications to Algebra and Geometry, and Analytical Mechanics, Bartholomew Price
A treatise on infinitesimal calculus, Bartholomew Price
AuteurBartholomew Price
ÉditeurThe University Press, 1856
Original provenant deUniversité d'Oxford
Numérisé13 sept. 2007
  
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