and for unit of length mv3
Hence if T-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 / feet and its own mass w pounds. Then T= Wg, lm = w, and the velocity of the undulation is
(j) 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, x, from the mean, on one side, leads to a rise, x, on the other; and if the whole column of fluid be of length 2a, we have the mass 2aSp disturbed through a space x, and acted on by a force 2Sxgp tending to bring it back. The time of oscillation is therefore (§ (a)) 2π and is the same for all liquids whatever be their densities.
Closed surface, ff 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 427, nearly opposed 428, at right angles 429, of any set of forces acting on a rigid body 570; Couples in same plane 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 g. Compressibility 651
Conditions of equilibrium of a particle 408; a material point 470; of parallel forces 558; of floating bodies 702-9; of any number of couples 567 et seq. Cone, orthogonal and oblique section of very small 486; solid angle of 482; area of segment cut from spherical surface by a small cone 487 Cones opposite or vertical 481
Confocal ellipsoids, corresponding points on 535; Chasles' proposition 537 Conical pendulum, Appendixƒ
Conical surface 480
Conservation of energy 250 Conservative system 243 Constancy of a balance 384
Constraint of a point 165, of a body 167; one degree of constraint of the most general character 170 Contrary forces 555 note Continuity, equation of 162 Conversion of units:-pounds per sq. inch to grammes per sq. centimetre 661; other units 362-366 Co-ordinates 452; propositions in co- ordinate geometry 459 Cord round cylinder 592, 603 Corresponding points in confocal ellip- soids 535
Cosines, sum of the squares of the direc- tion, of a line, equal to unity 460 Couple 201, axis of 201, moment of 201, direction of 560 -composition of in same or parallel planes 561; any number 564; any number not in parallel planes 566; conditions of equilibrium of 567; and a force, composition of 568 et seq. Curvature of a plane curve 9; integral 14; average 14; of a surface 120 of oblique sections, Meunier's Theorem, 121; principal, Euler's Theorem 122 Curvature of a lens, how to measure 381 Curve, plane 11; tortuous 13; of double
curvature II; continuous 35; closed 443
Curves use of, in representing experi mental results 347
Cycloid 66, 103; properties of 104; prolate 103; curtate io3
D'Alembert's Principle 230 Day, Sidereal and Mean Solar 357 Degrees of freedom and constraint 165, of a point 165. of a body 167; one degree of freedom of most general character 170
Density 174; linear, surface, volume,
477; mean density of the earth ex- pressed in attraction units 715 Developable surface 125; practical con- struction of a, from its edge 133 Diagonal scale 372
Direction of motion 8
Direction of rotation, positive 455 Direction cosine 463; sum of squares
of, equal to unity 460; of the common perpendicular to two lines 464 Displacement of a plane figure in its
solid equilibrium of 667
wire or fibre 605
Elasticity, co-efficient of 265 note; of volume 651; of figure 651
Electric images 528; definition 530; transformation by reciprocal radius vectors 531; electric image of a straight line, an angle, a circle, a sphere, a plane 531; application to the potential 532; of any distribution of attracting matter on a spherical shell 533; uniform shell eccentrically re- flected 533; uniform solid sphere eccentrically reflected 534 Elements of a force 184 Ellipse, how to draw an 19 Ellipsoid, central 237 Ellipsoid, attraction of a, 535; corre- sponding points on two 535.; Ellips- oidal shell defined 535; attraction of homogeneous ellipsoidal shell on in- ternal point 536; Potential constant inside 536; Chasles' Proposition con- cerning 537; equipotential surfaces of a 537; Maclaurin's Theorem 539; Ivory's 540; comparison of the po- tentials of two 537
Ellipsoid, Strain 141; principal axes of 142
Empirical formulae, use of 350 Energy, kinetic 179; kinetic energy of a system 234; energy in abstract dynamics 241, 251; foundation of the theory of energy 244; potential energy of a conservative system 245; conservation of E. 250; inevitable loss
of energy of visible motion 247; po- tential energy of a perfectly elastic body strained 644; energy of a strained isotropic substance 666 Epicycloid, integral curvature of 14, motion in 105
Epoch in simple harmonic motion 71 Equation of continuity 162; integral and differential 163
Equations of motion of any system 258 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 601. 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 122, on Impact 276 Evolute 20, 22
Flexible and inextensible line, Kine- 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 605; laws of flexure
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 614; 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 Fluid
Fluid, properties of perfect 401, 684; fluid pressure 685, equal in all di- rections 686, proved by energy cri 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 693; rate of increase of pressure 694, in a calm atmosphere of uniform tem. perature 695 (free surface in open vessel is level 696); resultant pres sure on a plane area 703; moment of pressure 702; 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 spheroid is a figure of equilibrium of a rotating incompressible homoge- neous 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
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 grav 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 476; representation of forces by lines 192; component of force 193; composition
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