Page images
PDF
EPUB
[ocr errors]

Wig

and for unit of length mo. Hence if 7=mv the theorem is true. If we suppose a portion of the tube to be straight, and the whole to be moving with velocity v parallel to this line, and against the motion of the cord, we shall have the straight part of the cord reduced to 'rest, and an undulation, of any, but unvarying, form and dimensions,

T running along it with the linear velocity

Suppose the cord stretched by an appended mass of W pounds, and suppose its length I feet and its own mass w pounds. Then I = Wg, Im=w, and the velocity of the undulation is

feet per second. (0) When an incompressible liquid escapes from an orifice, the velocity is the same as would be acquired by falling from the free surface to the level of the orifice.

For, as we may neglect (provided the vessel is large compared with the orifice) the kinetic energy of the bulk of the liquid; the kinetic energy of the escaping liquid is due to the loss of potential energy of the whole by the depression of the free surface. Thus the proposition at once.

(k) The small oscillations of a liquid in a U tube follow the harmonic law.

The tube being of uniform section S, a depression of level, 4, from the mean, on one side, leads to a rise, x, on the other; and if the whole column of fuid be of length på, we have the mass 2a Sp disturbed through a space x, and acted on by a force 2 Sxgp tending to bring it back. The time of oscillation is therefore ($ (a)) 21 and is the same for all liquids whatever be their densities.

[ocr errors]

INDEX.

494 (6)

Aberration gives hodograph of Earth's

orbit 53
Abscissae 452
Absolute acceleration 64

motion 63
- unit of force, Gauss's, 188; British 190
Acceleration, definition 34; uniform 32;

variable 33; average 33; angular 57;
composition and resolution 34, 37
- directed to a fixed centre 45

in a fixed direction 44
in logarithmic spiral with uniform

angular velocity about the pole 295
- in Şimple Harmonic Motion 74

in straight line, uniform 43
in uniform circular motion 36, 39, 42
-of momentum 178
Accurate measurements, necessity for

352
Action, Least 279

Maximum 317
Minimum 311
Stationary 281

Varying 282
Aeolotropic substance, an 647
Alteration of latitude by hemispherical

hill, or cavity 496; by a crevasse 497
Ampère's Theory of Electrodynamics

336
Amplitude of S. H. M. 71
Angle between two lines, definition of
Angle of repose 473
Angle, solid 482; round a point 483 ;

subtended at a point 485
Angular acceleration 57
Angular measure, standard of 357
Angular velocity 54; unit of 55; com-

position of 107, 108..
Anticlastic surface 120
Approximate' treatment of physical

questions 391
Arc, definition of 438; projection of
Area of an autotomic plane circuit 445
Argument of S. H. M. 71
Atmosphere Homogeneous 695 ; see

Homogeneous

Attraction not modified by interposition
of other matter 474

is normal to equipotential surfaces
506
- integral of normal, over a closed
surface 510

direct analytical calculation of 494
- law of, when a uniform spherical
shell exerts no attraction on an in.
ternal point 541
law of gravitation 475
of gravitating, electric, or magnetic
masses 478
- variation of, in crossing an attracting
surface 495

of a circular arc for a particle at its
centre 499

of a right cone for a particle at its
vertex 494 (c)
of a cylinder on a particle in its axis
of a cylindrical distribution of matter
508
of a uniform circular disc on a par.

ticle in its axis 494 (a)
- of an infinite disc 494

of two equal uniform discs, one posi-
tive, other negative 494 (d)

of an Ellipsoid 535, 537; of homo
geneous ellipsoid 538; Maclaurin's
Theorem 539; Ivory's Theorem 540;
Duhamel's application of Ivory's The.
orem 541
- of an ellipsoidal shell 535; on an in-
ternal particle 536

of a uniform limited straight line on
an external particle 499 (6)
of a mountain on a plumb line
496 (a)
at the top and the bottom of a pit
496 (6)
of infinite parallel planes 508
of a sphere composed of concentric
shells of uniform density 498
- of a uniform sphere on an external

particle infinitely near its surface 488
- of a uniform sphere $34. 541,

441 note

an 439

[ocr errors]

tion 541

of a 384

Attraction of an uninsulated sphere

under the influence of an electrified

particle 493
of a uniform spherical shell on an

internal point 479; converse proposi.
- of a uniform spherical shell on an

external point 488
- of a spherical surface whose density
varies as D-8 from excentric points
490 et seq.; excentric point inside at.
tracted point outside, and vice versa
491; excentric and attracted point

both within or both without 492
Autotomic circuit 443
'Average curvature 14
Average stress 645
Average velocity 26
Axiom, physical 209; regarding the

equilibrium of a non-rigid body

584
Axis of a couple 201
Axis, central 579
Balance, Coulomb's Torsion 385

- requisites for a good 383
-- sensibility, stability and constancy

statical principles of 592
Balance, spring 386
Ballistic pendulum 363, 273
Bending of a supported beam or uniform

bar 625; supported at ends or middle

625; at ends and middle 627
Bending, effect of, on cross section of

body 679
Bifilar suspension 388
Body, motion of a rigid 106
Body, a perfectly rigid, defined 393,

401
Bramah Press, hydrostatic principle of

689
British system of units of mass 661 note
British absolute unit of force 190
Cardioid 105
Catenary 594; a parabola 599; kinetic

question relative to 598; inverse pro-

blem 600
Cathetometer 38%
Central axis 579
Central ellipsoid 238
Centre of gravity, and centre of inertia

195, 542, 582; centrobaric bodies
542; if it exist is centre of inertia 550;
position of in stable equilibrium 585,
in rocking stones 586; of a body in
equilibrium about an axis 587, on a
fixed surface 588; Pappus' theorem
concerning 589

Centre of pressure 688, 703
Centre of mass or inertia 195, 583;

motion of centre of inertia of a rigid
body 232, 550; moments of inertia
of centrobaric body, round axes

through centre of inertia 551
Centrobaric body 542, proved possible

by Green 543, properties of 545;
centrobaric shell 547; centrobaric
solid 549; moments of inertia of a
centrobaric body round axes through
centre of inertia 551
Change of velocity 177,- of momen.

tum 177
Characteristic function, Hamilton's 283
Chasles on confocal ellipsoids 537
Chronometer 367
Chronoscope 369
Circuit, linear 443; autotomic 443
Circular measure, unit. of 357
Clairault's formula for the amount of

gravity at a place 187
Clocks 367
Closed curve 443

- polygon 443
Closed surface, si Ndo, over a 510
Coarsegrainedness 646
Coefficient of elasticity 265 note, 644
Coefficient of restitution 265; of glass,

iron, wool 265
Comet, hodograph of orbit of 49
Component velocity 29; acceleration

37; of a force, effective 193
Composition of Velocities 31; Accelera.

tions 34; Simple Harmonic Motions in
same direction 75, in different direc.
tions 80; Angular velocities 107,
about axes meeting in a point 108;
Rotations 107, successive finite rota-
tions 109; Forces 221, of two acting
on a point 419, 422, special cases of
423 et seq.; nearly conspiring 4?7,
nearly opposed 428, at right angles
429, of any set of forces acting on a
rigid body 570; Couples in same planer
or in parallel planes 561, 562, 563, any
number, 564; not in parallel planes
565, any number of 566, and a force

568
Compound pendulum, Appendix gi
Compressibility 651
Conditions of equilibrium of a particle

408; a material point 470; of parallel
forces 558; of floating bodies 703-9;

of any number of couples 567 et seq.
Cone, orthogonal and oblique section of

very small 486; solid angle of 481;
area of segment cut from spherical

surface by a small cone 487
Cones opposite or vertical 481

293, 683

rem $41

soids 535

Confocalellipsoids, corresponding points plane 91, examples 36; of a rigid
on 535; Chasles' proposition 537

solid 100
Copical pendulum, Appendix f

Dissipation of Energy, instances 347,
Conical surface 480
Conservation of energy 250

Dissipative systems' 292
Conservative system 343

Distortion, places of maximum, in a
Constancy of a balance 384

cylinder 678
Constraint of a point 165, of a body Distribution of the weight of a solid on

167; one degree of constraint of the points supporting it 636
most general character 170

Double-weighing 384
Contrary forces 555 note

Dubamel's application of Ivory's theo.
Continuity, equation of 161
Conversion of units :-pounds per sq. Dynamics :

inch to grammes per sq. centimetre

661; other units 362-366
Co-ordinates 452; propositions in co-

Edge of regression 139,
ordinate geometry 459

Elastic body, a perfectly 643
Cord round cylinder 592, 603

centre of a section of a wire 608;
Corresponding points in confocal ellip line of elastic centres, 608; rotation

of a wire about 628
Cosines, sum of the squares of the direc - curve transmitting force and couple

tion, of a line, equal to unity 460 619, properties of 620; Kirchoff's ki.
Couple 201, axis of 201, moment of betic comparison, common pendulum
201, direction of 560

and elastic curve 620
- composition of in same or parallel solid equilibrium of 667
planes 561; any number 564; any wire or fibre 60s
number not in parallel planes 566 ; Elasticity, co-efficient of 263 note; of
conditions of equilibrium of 567; volume 651; of figure 651

and a force, composition of 568 et seq: Electric images 528; definition 530;
Curvature of a plane curve 9; integral transformation by reciprocal radius

14; average 14; of a surface 120 of vectors 531; electric image of a
oblique sections, Meunier's Theorem, straight line, an angle, a circle, &

121; principal, Euler's Theorem 132 sphere, a plane 531; application to the
Curvature of a lens, how to measure 381 potential 532; of any distribution of
Curve, plane 11; tortuous 13; of double attracting

matter on a spherical shell
curvature it; continuous 35; closed 533; uniform shell eccentrically re-
443

Hected 533; uniform solid sphere
Curves use of, in representing experi eccentrically reflected 534
mental results 347

Elements of a force 184
Cycloid 66, 103 ; properties of 104; Ellipse, how to draw an 19
prolate 103; curtate io3

Ellipsoid, central 337

Ellipsoid, attraction of a, 533; corre.
D'Alembert's Principle 330

sponding points on two 535.; Ellips.
Day, Sidereal and Mean Solar 357

oidal shell defined 535; attraction of
Degrees of freedom and constraint 165, homogeneous ellipsoidal shell on in.

of a point 165. of a body 167; one ternal point 536; Potential constant
degree of freedom of most general inside 536; Chasles' Proposition con
character 170

cerning 537: equipotential surfaces of
Density 174; linear, surface, volume, & 537; Maclaurin's Theorem 539;

477 ; mean density of the earth ex Ivory's 540; comparison of the po
pressed in attraction units 715

tentials of two 537
Developable surface 135; practical COD Ellipsoid, Strain 141; principal axes

struction of a, from its edge 133
Diagonal scale 372

Empirical formulae, use of 350
Direction of motion 8

Energy, kinetic 179; kinetic energy of
Direction of rotation, positive 455

& system 334; energy in abstract
Direction cosine 463; sum of squares dynamics 341, 351; foundation of

of, equal to unity 460; of the common the theory of energy 244; potential
perpendicular to two lines 464

energy of a conservative system 145;
Displacement of a plane figure in its conservation of E. ago; inevitable love

of 147

of energy of visible motion 347; po-
tential energy of a perfectly elastic
body strained 644; energy of a

strained isotropic substance 666
Epicycloid, integral curvature of 340

motion in 105
Epoch in simple harmonic motion 71
Equation of continuity 162; integral and

differential 163
Equations of motion of any system 358
Equilibrium of a particle, conditions of

408, 470, on smooth and rough curves
and surfaces 473; conditions of equi.
librium of forces acting at a point 470;
conditions of equilibrium of three forces
acting at a point 584; graphic test
of forces in equilibrium 414; condi.
tions for stable equilibrium of a body
585, rocking stones 586, body move-
able about an axis 587, body on a
fixed surface 588; neutral, stable, and
unstable equilibrium, tested by the
principle of virtual velocities 256,
energy criterion of 257; conditions of
equilibrium of parallel forces 558; con.
ditions of equilibrium of forces acting
on a rigid body 576; equilibrium of a
non-rigid body not affected by ad.
ditional fixtures 584, of a flexible and
inextensible cord 594; position of equi.
librium of a flexible string on a smooth
surface bor, rough surfaces 602; equi.
librium of elastic solid 667, of incom.
pressible fluid completely filling rigid
vessel 696, under any system of forces
697; equilibrium of a floating body 704

et seq., of a revolving mass of fluid 710
Equipotential surfaces, examples of 499,

505, 526, of ellipsoidal shell 537
Equivalent of pounds per square inch

in grammes per square centimetre

661; other units 362-366.
Ergometer 389, Morin's 389
Experience 320
Experiment and observation 324; rules

for the conduct of experiment 325; use
of empirical formulae in exhibiting

results of experiment 347
Euler's theorem on curvature 111, OD

Impact 276
Evolute 20, 23
Flexible and inextensible line, Kinc-

matics of a 16; flexible and inexten.
sible surface, flexure of 125, general
property of 134; flexible string on
smooth surface, position of equili.
brium of 601, on rough surface 602
Flexure of flexible and inextensible sur-

face 125, of a wire 60;; laws of Aesure

and torsion 607; axes of pure flexure
609; case in which the elastic central
line is a normal axis of torsion 609 ;
where equal flexibility in all directions
610; wire strained to any given spiral
and twist 612; spiral spring 814;
principal axes of 679; distortion of the
cross section of a bent rod 679
Floating bodies, stable equilibrium of

lemma 704; stability of 705 et seq.;
see Florid
Fluid, properties of perfect 401, 684;
Auid pressure 685, equal in all di-
rections 686, proved by energy crie
terion 689; fluid pressure as depending
on external forces 690; surfaces of
equal pressure are perpendicular to
lines of force 691, are surfaces of
equal density and equal potential 69? ;
rate of increase of pressure 694. in
a calm atmosphere of uniform tem-
perature 695 (free surface in open
vessel is level 696); resultant preso
sure on a plane area 703; moment of
pressure 703; loss of apparent weight
by immersion 703; conditions of equi.
librium of a fluid completely filling a
closed vessel 696, under non-con-
servative system of forces 697, im-
aginary example 699, actual case
701; equilibrium of a floating body,
lemma 704, stability 705, work done
in a displacement 705, metacentre,
condition of its existence 709; oblate
spberoid is a figure of equilibrium of
a rotating incompressible homoge-
aeous fluid mass 711; relation be-
tween angular velocity of rotation
and density with given ellipticity 712;
table of eccentricities and correspond.
ing angular velocities and moments
of momentum for a liquid of the
earth's mean density 717 ; equilibrium
of rotating ellipsoid of three unequal

axes 719
Fluxion 38
Forbes's use of Viscous in connection

with glacier motion 683
Force, moment of 46, about a point

199, source of the idea of 173, de.
fined 183, specification of a 184,
measure of a 185, measurement of
224, by pendulum 387; force of gram
ity, Clairault's formula for 187, in
absolute units 187, average, in Britain
191 ; unit of force, gravitation 185,
absolute 188; British absolute unit
191; attraction unit of force 176;
representation of forces by lines 192;
component of force 193; composition

« PreviousContinue »