Velocity Sentence Examples

The wind velocity did not exceed 20 km.

The concentration is known, and the conductivity can be measured experimentally; thus the average velocity with which the ions move past each other under the existent electromotive force can be estimated.

When he considered all days irrespective of wind velocity, Mazelle found the influence of temperature obliterated.

After Professor Amund Helfand had, in July 1875, discovered the amazingly great velocity, up to 644 ft.

The velocity of light (q.v.) has been measured with all the precision necessary for the purpose.

Hence the elementary arc divided by the element of time is the rate of change of velocity of the moving-point, or in other words, the velocity in the hodograph is the acceleration in the orbit.

Rising in the high tablelands or on the slopes of the Drakensberg or Lebombo mountains the rivers in their upper courses have a great slope and a high velocity.

In this way the medium velocity of the current may be diminished, and consequently the quantity of water discharged in a given time must, from the effects of friction, be considerably less than that which is computed from theory.

From a collection of the best experiments by previous workers he selected eighty-two (fifty-one on the velocity of water in conduit pipes, and thirty-one on its velocity in open canals); and, discussing these on physical and mechanical principles, he succeeded in drawing up general formulae, which afforded a simple expression for the velocity of running water.

In particular, for a jet issuing into the atmosphere, where p=P, q 2 /2g = h - z, (9) or the velocity of the jet is due to the head k-z of the still free surface above the orifice; this is Torricelli's theorem (1643), the foundation of the science of hydrodynamics.

At the main base in Adelie Land autumn sledging proved impossible, and throughout the winter there was a continuous succession of terrific blizzards, wind with an average velocity of 50 m.p.h.

Simspon concluded that for a given wind velocity dissipation is practically a linear function of ionization.

The other forms of velocity anemometer may be described as belonging to the windmill type.

The second method is in principle extremely simple, consisting merely in multiplying the observed velocity of light by the time which it takes light to travel from the sun to the earth.

The velocity is now well determined; the difficulty is to determine the time of passage.

He supposed that the filaments of water which graze along the sides of the pipe lose a portion of their velocity; that the contiguous filaments, having on this account a greater velocity, rub upon the former, and suffer a diminution of their celerity; and that the other filaments are affected with similar retardations proportional to their distance from the axis of the pipe.

In the Eulerian method the attention is fixed on a particular point of space, and the change is observed there of pressure, density and velocity, which takes place during the motion; but in the Lagrangian method we follow up a particle of fluid and observe how it changes.

I n a straight uniform current of fluid of density p, flowing with velocity q, the flow in units of mass per second across a plane area A, placed in the current with the normal of the plane making an angle 0 with the velocity, is oAq cos 0, the product of the density p, the area A, and q cos 0 the component velocity normal to the plane.

Generally if S denotes any closed surface, fixed in the fluid, M the mass of the fluid inside it at any time t, and 0 the angle which the outward-drawn normal makes with the velocity q at that point, dM/dt = rate of increase of fluid inside the surface, (I) =flux across the surface into the interior _ - f f pq cos OdS, the integral equation of continuity.

A small sphere of the fluid, if frozen suddenly, would retain this angular velocity.

Calling the sum of the pressure and potential head the statical head, surfaces of constant statical and dynamical head intersect in lines on H, and the three surfaces touch where the velocity is stationary.

Thus if d,/ is the increase of 4, due to a displacement from P to P', and k is the component of velocity normal to PP', the flow across PP' is d4 = k.PP'; and taking PP' parallel to Ox, d,, = vdx; and similarly d/ ' = -udy with PP' parallel to Oy; and generally d4,/ds is the velocity across ds, in a direction turned through a right angle forward, against the clock.

The curves 0 = constant and 4, = constant form an orthogonal system; and the interchange of 0 and 4, will give a new state of uniplanar motion, in which the velocity at every point is turned through a right angle without alteration of magnitude.

For instance, in a uniplanar flow, radially inward towards 0, the flow across any circle of radius r being the same and denoted by 27rm, the velocity must be mfr, and 0=m log r,, y=m0, +4,i =m log re ie, w=m log z.

A single vortex will remain at rest, and cause a velocity at any point inversely as the distance from the axis and perpendicular to its direction; analogous to the magnetic field of a straight electric current.

If other vortices are present, any one may be supposed to move with the velocity due to the others, the resultant stream function being = gy m log r =log IIrm; (9) the path of a vortex is obtained by equating the value of 1P at the vortex to a constant, omitting the rm of the vortex itself.

Uniplanar Motion of a Liquid due to the Passage of a Cylinder through it.-A stream-function 4, must be determined to satisfy the conditions v24 =o, throughout the liquid; (I) I =constant, over any fixed boundary; (2) d,t/ds = normal velocity reversed over a solid boundary, (3) so that, if the solid is moving with velocity U in the direction Ox, d4y1ds=-Udy/ds, or 0 +Uy =constant over the moving cylinder; and 4,+Uy=41' is the stream function of the relative motion of the liquid past the cylinder, and similarly 4,-Vx for the component velocity V along Oy; and generally 1,1'= +Uy -Vx (4) is the relative stream-function, constant over a solid boundary moving with components U and V of velocity.

If the liquid is stirred up by the rotation R of a cylindrical body, d4lds = normal velocity reversed dy = - Rx- Ry ds (5) ds 4' + 2 R (x2 + y2) = Y, (6) a constant over the boundary; and 4,' is the current-function of the relative motion past the cylinder, but now V 2 4,'+2R =o, (7) throughout the liquid.

Over a concentric cylinder, external or internal, of radius r=b, 4,'=4,+ Uly =[U I - + Ui]y, (4) and 4" is zero if U 1 /U = (a 2 - b2)/b 2; (5) so that the cylinder may swim for an instant in the liquid without distortion, with this velocity Ui; and w in (I) will give the liquid motion in the interspace between the fixed cylinder r =a and the concentric cylinder r=b, moving with velocity U1.

If the liquid is reduced to rest at infinity by the superposition of an opposite stream given by w = - Uz, we are left with w = Ua2/z, (6) =U(a 2 /r) cos 0= Ua2x/(x2+y2), (7) 4, = -U(a 2 /r) sin 0= -Ua2y/( x2+y2), (8) giving the motion due to the passage of the cylinder r=a with velocity U through the origin 0 in the direction Ox.

If the direction of motion makes an angle 0' with Ox, tan B' = d0 !dam _ ?xy 2 = tan 20, 0 =-10', (9) dy/ y and the velocity is Ua2/r2.

When the cylinder r =a is moved with velocity U and r =b with velocity U 1 along Ox, = U b e - a,1 r +0 cos 0 - U ib2 - 2 a, (r +Q 2 ') cos 0, = - U be a2 a2 (b 2 - r) sin 0 - Uib2 b1)a, (r - ¢2 sin 0; b and similarly, with velocity components V and V 1 along Oy a 2 b2 ?= Vb,_a,(r+r) sin g -Vi b, b2 a, (r+ 2) sin 0, (17) = V b, a2 a, (b2 r) cos 0+Vi b, b, a, (r- ¢ 2) cos h; (18) and then for the resultant motion z 2zz w= (U 2 + V2)b2a a2U+Vi +b a b a2 U z Vi -(U12+V12) b2 z a2b2 Ui +VIi b 2 - a 2 U1 +Vii b 2 - a 2 z The resultant impulse of the liquid on the cylinder is given by the component, over r=a (§ 36), X =f p4 cos 0.ad0 =7rpa 2 (U b z 2 + a 2 Uib.2bz a2); (20) and over r =b Xi= fp?

With v=o, the angular velocity of the cylinder is 2w; in this way the velocity may be calculated of the propagation of ripples and waves on the surface of a vertical whirlpool in a sink.

Another explanation may be given of the sidelong force, arising from the velocity of liquid past a cylinder, which is encircled by a vortex.

The resultant hydrostatic thrust across any diametral plane of the cylinder will be modified, but the only term in the loss of head which exerts a resultant thrust on the whole cylinder is 2mU sin Olga, and its thrust is 27rpmU absolute units in the direction Cy, to be counteracted by a support at the centre C; the liquid is streaming past r=a with velocity U reversed, and the cylinder is surrounded by a vortex.

The velocity of a liquid particle is thus (a 2 - b 2)/(a 2 +b 2) of what it would be if the liquid was frozen and rotating bodily with the ellipse; and so the effective angular inertia of the liquid is (a 2 -b 2) 2 /(a 2 +b 2) 2 of the solid; and the effective radius of gyration, solid and liquid, is given by k 2 = 4 (a 2 2), and 4 (a 2 For the liquid in the interspace between a and n, m ch 2(0-a) sin 2E 4) 1 4Rc 2 sh 2n sin 2E (a2_ b2)I(a2+ b2) = I/th 2 (na)th 2n; (8) and the effective k 2 of the liquid is reduced to 4c 2 /th 2 (n-a)sh 2n, (9) which becomes 4c 2 /sh 2n = s (a 2 - b 2)/ab, when a =00, and the liquid surrounds the ellipse n to infinity.

An angular velocity R, which gives components - Ry, Ix of velocity to a body, can be resolved into two shearing velocities, -R parallel to Ox, and R parallel to Oy; and then ik is resolved into 4'1+1'2, such that 4/ 1 -R-Rx 2 and 1//2+IRy2 is constant over the boundary.

In a similar way the more general state of motion may be analysed, given by w =r ch2('-y), y =a+, i, (26) as giving a homogeneous strain velocity to the confocal system; to which may be added a circulation, represented by an additional term in w.

Motion symmetrical about an Axis.-When the motion of a liquid is the same for any plane passing through Ox, and lies in the plane, a function ' can be found analogous to that employed in plane motion, such that the flux across the surface generated by the revolution of any curve AP from A to P is the same, and represented by 2s-4 -11'o); and, as before, if d is the increase in due to a displacement of P to P', then k the component of velocity normal to the surface swept out by PP' is such that 274=2.7ryk.PP'; and taking PP' parallel to Oy and Ox, u= -d/ydy, v=dl,t'/ydx, (I) and 1P is called after the inventor, " Stokes's stream or current function," as it is constant along a stream line (Trans.

The vortex advances with a certain velocity; and if an equal circular vortex is generated coaxially with the first, the mutual influence can be observed.

The components of velocity of the moving origin are denoted by U, V, W, and the components of angular velocity of the frame of reference by P, Q, R; and then if u, v, w denote the components of fluid velocity in space, and u', v', w' the components relative to the axes at a point (x, y, z) fixed to the frame of reference, we have u =U +u' - yR +zQ, v =V +v -zP +xR, w=W +w -xQ +yP.

Thus, for example, with = 4Uy 2 (r 2 a 2 -I), r2 = x2 +y 2, (13) for the space inside the sphere r=a, compared with the value of, i' in § 34 (13) for the space outside, there is no discontinuity of the velocity in crossing the surface.

Hill's spherical vortex, advancing through the surrounding liquid with uniform velocity.

As an application of moving axes, consider the motion of liquid filling the ellipsoidal case 2 y 2 z2 Ti + b1 +- 2 = I; (1) and first suppose the liquid be frozen, and the ellipsoid l3 (4) (I) (6) (9) (I o) (II) (12) (14) = 2 U ¢ 2, (15) rotating about the centre with components of angular velocity, 7 7, f'; then u= - y i +z'i, v = w = -x7 7 +y (2) Now suppose the liquid to be melted, and additional components of angular velocity S21, 522, S23 communicated to the ellipsoidal case; the additional velocity communicated to the liquid will be due to a velocity-function 2224_ - S2 b c 6 a 5 x b2xy, as may be verified by considering one term at a time.

To determine the motion of a jet which issues from a vessel with plane walls, the vector I must be constructed so as to have a constant (to) (II) the liquid (15) 2, integrals;, (29) (30) (I) direction 0 along a plane boundary, and to give a constant skin velocity over the surface of a jet, where the pressure is constant.

The stream lines xBAJ, xA'J' are given by = 0, m; so that if c denotes the ultimate breadth JJ' of the jet, where the velocity may be supposed uniform and equal to the skin velocity Q, m=Qc, c=m/Q.

Ja - u  ?I a -a b -u' sh nS2=sh log (Q)=?a - b a - a' b - u' At x where = co, u = o, and q= go, (O n b - a ' a + a -b a' cio) - ?a-a'?b a-a' q In crossing to the line of flow x'A'P'J', b changes from o to m, so that with q = Q across JJ', while across xx the velocity is qo, so that i n = go.

The motion of a jet impinging on an infinite barrier is obtained by putting j = a, j' = a'; duplicated on the other side of the barrier, the motion reversed will represent the direct collision of two jets of unequal breadth and equal velocity.

The continuity is secured if the liquid between two ellipsoids X and X 11 moving with the velocity U and 15 1 of equation (II), is squeezed out or sucked in across the plane x=o at a rate equal to the integral flow of the velocity I across the annular area a l.

When the liquid is bounded externally by the fixed ellipsoid A = A I, a slight extension will give the velocity function 4 of the liquid in the interspace as the ellipsoid A=o is passing with velocity U through the confocal position; 4 must now take the formx(1'+N), and will satisfy the conditions in the shape CM abcdX ¢ = Ux - Ux a b x 2+X)P Bo+CoB I - C 1 (A 1 abcdX, I a1b1cl - J o (a2+ A)P and any'confocal ellipsoid defined by A, internal or external to A=A 1, may be supposed to swim with the liquid for an instant, without distortion or rotation, with velocity along Ox BA+CA-B 1 -C1 W'.

A distribution of sources and doublets over a moving surface will enable an expression to be obtained for the velocity function of a body moving in the presence of a fixed sphere, or inside it.

The partial differential coefficient of T with respect to a component of velocity, linear or angular, will be the component of momentum, linear or angular, which corresponds.

Conversely, if the kinetic energy T is expressed as a quadratic function of x, x x3, y1, y2, y3, the components of momentum, the partial differential coefficient with respect to a momentum component will give the component of velocity to correspond.

These theorems, which hold for the motion of a single rigid body, are true generally for a flexible system, such as considered here for a liquid, with one or more rigid bodies swimming in it; and they express the statement that the work done by an impulse is the product of the impulse and the arithmetic mean of the initial and final velocity; so that the kinetic energy is the work done by the impulse in starting the motion from rest.

It is driven by a powerful engine through triple gearing of 42 to 1, and speeded to have a surface velocity of rollers of 15 ft.

By the mode of admission the hot liquor at its entry is distributed over a large area relatively to its volume, and while this is necessarily effected with but little disturbance to the contents of the vessel, a very slow velocity is ensured for the current of ascending juice.

The slower class of meteors overtaking the earth (like the Andromedids of November) have a velocity of about 8 or 10 m.

The meteors move very slowly, as they have to overtake the earth, and their apparent velocity is only about 9 m.

Similar condensations produced the sun and stars; and the flaming state of these bodies is due to the velocity of their motions.

A comparison of the force habitually developed by the wind in various parts of the islands shows that at Suttsu in Yezo the average strength is 9 metres per second, while Izuhara in the island Tsu-shima, Kumamoto in KiOshi and Gifu in the east centre of the main island stand at the bottom of the list with an average wind velocity of only 2 metres.

Their season is from June to October, but they occur in other months also, and they develop a velocity of 5 to 75 m.

An exactly similar expression holds good in hydrokinetics, provided that for the electric potential we substitute velocity potential, and for the electric force the velocity of the liquid.

These are paved with stone blocks or lined with mercury riffles, so that from the greatly reduced velocity of flow, due to the sudden increase of surface, the finer particles of gold may collect.

In 1628 Castelli published a small work, Della misura dell' acque correnti, in which he satisfactorily explained several phenomena in the motion of fluids in rivers and canals; but he committed a great paralogism in supposing the velocity of the water proportional to the depth of the orifice below the surface of the vessel.

This increase of velocity implies an increase of the reaction on the surface, the black side of a vane being thus pressed with greater force than the bright side.

In later memoirs Reynolds followed up this subject by proceeding to establish definitions of the velocity and the momentum and the energy at an element of volume of the molecular medium, with the precision necessary in order that the dynamical equations of the medium in bulk, based in the usual manner on these quantities alone, without directly considering thermal stresses, shall be strictly valid - a discussion in which the relation of ordinary molar mechanics to the more complete molecular theory is involved.

He had learnt from Torstensson that Denmark was most vulnerable if attacked from the south, and, imitating the strategy of his master, he fell upon her with a velocity which paralysed resistance.

To remedy drawback (2) Repsolds provided for the Yale heliometer an additional handle for motion in position angle, intermediate in velocity between the original quick and slow motions.

The difference between the position as determined astronomically and by dead-reckoning gives an excellent idea of the general direction and velocity of the surface currents.

This deflecting force is directly proportional to the velocity and the mass of the particle and also to the sine of the latitude; hence it is zero at the equator and comes to a maximum at the poles.

The deeper layers lag behind the upper in deflection and the velocity of the current rapidly diminishes in consequence.

He called the depth at which the opposed direction is attained the driftcurrent depth, and he found it to be dependent on the velocity of the surface current and on the latitude.

By the use of the spiral guide casing and the chimney the velocity of the effluent air is gradually FIG.

The angular velocity of the shaft is proportional to the rate of working.

A number of molecules moving in obedience to dynamical laws will pass through a series of configurations which can be theoretically determined as soon as the structure of each molecule and the initial position and velocity of every part of it are known.

Each impinging molecule exerts an impulsive pressure equal to mu on the boundary before the component of velocity of its centre of gravity normal to the boundary is reduced to zero.

A particle of this mass is easily visible microscopically, and a velocity of 2 mm.

But it can be shown that from the aggregation of these separate short motions the particle ought to have a resultant motion, described with an average velocity which, although much smaller than 2 mm.

Modern research has proved that such reactions are not occasioned by water acting as H 2 0, but really by its ions (hydrions and hydroxidions), for the velocity is proportional (in accordance with the law of chemical mass action) to the concentration of these ions.

It passes over equal spaces in equal times, but falls with an accelerating velocity according to the formula h = zgt 2, where h is the height fallen through, g the force of gravity, and t the time of flight.

The yard scales were on detachable strips, so that fresh strips could be inserted for variations in velocity.

C. Vogel's spectroscopic measures in 1889.2 Previously to each obscuration, the star was found to be moving rapidly away from the earth; its velocity then diminished to zero pari passu with the loss of light, and reversed its direction during the process of recovery.

If each wave travels out from the source with velocity U the n waves emitted in one second must occupy a length U and therefore U = nX.

The distribution of velocity then is represented by the dotted curve and is forward when the curve is above the axis and Dackward when it is below.

To find the relation of the velocity to displacement and pressure we shall express the fact that the wave travels on carrying all its conditions with it, so that the displacement now at M will arrive at N while the wave travels over MN.

Then u/U = - dy/dx (2) This gives the velocity of any particle in terms of the displacement.

Equating (I) and (2) u/U = wÆ (3) which gives the particle velocity in terms of the pressure excess.

Generally, if any condition in the wave is carried forward unchanged with velocity U, the change of 4 at a given point in time dt is equal to the change of as we go back along the curve a distance dx = Udt at the beginning of dt.

It is convenient to give this calculation before proceeding to describe the experimental determination of the velocity in air, in other gases and in water, since the calculation serves to some extent as a guide in conducting and interpreting the observations.

Every particle in the plane will have the same displacement and the same velocity, and these will be perpendicular to the plane and parallel to the line of propagation.

Whatever the form of a wave, we could always force it to travel on with that form unchanged, and with any velocity we chose, if we could apply any " external " force we liked to each particle, in addition to the " internal " force called into play by the compressions or extensions.

But it has velocity U, and therefore momentum poU 2 is carried in.

If the velocity of a particle at A relative to the undisturbed parts is u from left to right, the velocity of the matter moving out at A is U - u, and the momentum carried out by the moving matter is p(U - u) 2.

If then we apply a pressure X given by (5) at every point, and move the medium with any uniform velocity U, the disturbance remains fixed in space.

Or if we now keep the undisturbed parts of the medium fixed, the disturbance travels on with velocity U if we apply the pressure X at every point of the disturbance.

If the velocity U is so chosen that E - poU 2 = o, then X = o, or the wave travels on through the action of the internal forces only, unchanged in form and with velocity U = (E/p).

If we omitted it we should have to assume this, and equation (6) would give us the velocity of propagation if the assumption were justified.

If, however, we put on external forces of the required type X it is obvious that any wave can be propagated with any velocity, and our investigation shows that when U has the value in (6) then and only then X is zero everywhere, and the wave will be propagated with that velocity when once set going.

He supposed that in air Boyle's law holds in the extensions and compressions, or that p = kp, whence dp/dp = k = p/p. His value of the velocity in air is therefore U = iJ (p ip.) (Newton's formula).

But for very small times the assumption may perhaps be made, and the result at least shows the way in which the velocity is affected by the addition of a small term depending on and changing sign with u.

We see at once that, where u=o, the velocity has its " normal " value, while where u is positive the velocity is in excess, and where u is negative the velocity is in defect of the normal value.

In ordinary sound-waves the effect of the particle velocity in affecting the velocity of transmission must be very small.

The maximum particle velocity is 21rna (where n is the frequency and a the amplitude), or 27raU/X.

But there is no doubt that with very loud explosive sounds the normal velocity is quite considerably exceeded.

This is hardly to be explained by equation (I I), for at the very front of the disturbance u =o and the velocity should be normal.

The kinetic energy per cubic centimetre is 2 pu t, where is the density and u is the velocity of disturbance due to the passage of the wave.

An obvi us method of determining the velocity of sound in air consists in starting some sound, say by firing a gun, and stationing an observer at some measured distance from the gun.

The distance divided by the time gives the velocity of the sound.

Regnault in the years 1862 to 1866 on the velocity of sound in open air, in air in pipes and in various other gases in pipes, he sought to eliminate personal equaticn by dispensing with the human element in the observations, using electric receivers as observers.

On page 459 of the Memoire will be found a list of previous careful experiments on the velocity of sound.

The temperature of the air traversed and its humidity were observed, and the result was finally corrected to the velocity in dry air at o C. by means of equation (ro).

In the memoir cited above Regnault gives an account of determinations of the velocity in air in pipes of great length and of diameters ranging from o 108 metres to i i metres.

He found that in all cases the velocity decreased with a diameter.

The sound travelled to and fro in the pipes several times before the signals died away, and he found that the velocity decreased with the intensity, tending to a limit for very feeble sounds, the limit being the same whatever the source.

He found that within wide limits the velocity was independent of the pressure, thus confirming the theory.

Correcting the velocity obtained in the 0 .

They found that the velocity of propagation of different musical sounds was the same.

When a wave of sound meets a surface separating two media it is in part reflected, travelling back from the surface into the first medium again with the velocity with which it approached.

When a wave of sound travelling through one medium meets a second medium of a different kind, the vibrations of its own particles are communicated to the particles of the new medium, so that a wave is excited in the latter, and is propagated through it with a velocity dependent on the density and elasticity of the second medium, and therefore differing in general from the previous velocity.

As with light the ratio involved in the second law is always equal to the ratio of the velocity of the wave in the first medium to the velocity in the second; in other words, the sines of the angles in question are directly proportional to the velocities.

Hence sound rays, in passing from one medium into another, are bent in towards the normal, or the reverse, according as the velocity of propagation in the former exceeds or falls short of that in the latter.

It further follows, as in the analogous case of light, that there is a certain angle termed the critical angle, whose sine is found by dividing the less by the greater velocity, such that all rays of sound meeting the surface separating two different bodies will not pass onward, but suffer total reflection back into the first body, if the.

On the other hand, to produce convergence with water or hydrogen gas, in both which the velocity of sound exceeds its rate in air, the lens ought to be concave.

Now if the temperature is higher overhead than at the surface, the velocity overhead is greater.

But usually the lower layers are warmer than the upper layers, and the velocity below is greater than the velocity above.

Stokes showed that this effect is one of refraction, due to variation of velocity of the air from the surface upwards Brit.

It is, of course, a matter of common observation that the wind increases in velocity from the surface upwards.

The velocity of any part of a wave front relative to the ground will be the normal velocity of sound + the velocity of the wind at that point.

Since the velocity increases as we go upwards the front tends to swing round and travel downwards, as shown in the successive positions a I, 2, 3 and 4, in fig.

But if the wind is against the sound the velocity of a point of the wave front is the normal velocity-the wind velocity at the point, and so decreases as we rise.

This will go on continually as long as air is supplied to the cylinder, and the velocity of rotation of the upper plate will be accelerated up to a certain maximum, at which it may be maintained by keeping the force of the current constant.

The result is a note whose pitch rises as the velocity of rotation increases, and becomes steady when that velocity reaches its constant value.

Since U=n X where U is the velocity of sound, X the wave-length, and n the frequency, it follows that the forward frequency is greater than the backward frequency.

Let S' be its position one second later, its velocity being u.

Let R be the receiver at a given instant, R' its position a second later, its velocity being v.

Let the velocity of the air from S to R be w, and let U be the velocity of sound in still air.

If now the curve moves along unchanged in form in the direction ABC with uniform velocity U, the epoch e =OA at any time t will be Ut, so that the value of y may be represented as 2 y=a sin T (x - Ut).

Then, as we shall prove later, the vibrations of the string may be represented by the travelling of two trains in opposite directions each with velocity /tension=mass per unit length each half the height of the train represented in fig.

The maximum velocity of a particle in the wave-train is the amplitude of dy/dt.

Since the velocity is the same for all disturbances they all travel at the same speed, and the two trains will always remain of the same form.

We see, then, that the conditions for the application of Fourier's theorem are equivalent to saying that all disturbances will travel along the system with the same velocity.

If U 4 is the velocity of longitudinal waves along the sounder, and 1 the length of the sounder, the frequency of vibration is U 8 /2l.

If L 1 is the internodal distance and U 1 the velocity in a gas, L and U being the corresponding values for air, we have U 1 /U =L1/L.

If U is the velocity of sound in a gas at pressure P with density p, and if waves of length X and frequency N are propagated through it, then the distanc?e l between the dust-heaps is 2 = N - zN Vyp' where y is the ratio of the two specific heats.

We shall first investigate the velocity with which a disturbance travels along a string of mass m per unit length when it is stretched with a constant tension T, the same at all points.

Then move AB from right to left with this velocity, and the disturbance remains fixed in space.

We shall find the velocity of propagation, just as in previous cases, from the consideration of transfer of momentum.

Suppose that a disturbance is travelling with velocity U unchanged in form along a rod from left to right.

At B there is only the latter kind, and since the transfer of matter is powoU, where po is the undisturbed density and wo is the undisturbed cross-section, since its velocity is U the passage of momentum per second is powoUo 2.

At A, if the velocity of the disturbance relative to undisturbed parts of the rod is u from left to right, the velocity relative to A is U - u.

The velocity with which the rod must travel in order that the disturbance may be fixed in space is therefore U =, I (Y/p), or, if the rod is kept fixed, this is the velocity with which the disturbance travels.

But keeping r/X small we may as before form stationary waves, and it is evident that the series of fundamental and overtones will be just as with the air in pipes, and we shall have the same three types - fixed at one end, free at both ends, fixed at both ends - with fundamental frequencies respectively 41, p ' 21 V p, and I velocity in rod =velocity in air X distance between dust heaps.

The velocity of propagation of a torsional disturbance along a wire of circular section may be found by the transfer of momentum method, remembering that we must now replace linear momentum by angular momentum.

If 0 is the angle of twist, the angular velocity is d0/dt.

The velocity of a disturbance along such a bar, and its modes of vibration, depend therefore on the elastic properties of the material and the dimensions of the bar.

Substituting in (33) we get U 2 = n/p. (34) If we now keep the wire at rest the disturbance travels along it with velocity U= d (nip), and it depends on the rigidity and density of the wire and not upon its radius.

When the velocity of the jet is gradually increased there is a certain range of velocity for which the jet is unstable, so that any deviation from the straight rush-out tends to increase as the jet moves up. If then the jet is just on the point of instability, and is subjected as its base to alternations of motion, the sinuosities impressed on the jet become larger and larger as it flows out, and the flame is as it were folded on itself.

But, if quite regular disturbances are impressed on the jet at intervals of time which depend on the diameter and speed of outflow (they must be somewhat more than ?r times its diameter apart), these disturbances go on growing and break the stream up into equal drops, which all move with the same velocity one after the other.

The third mode of production of combination tones, the production in the medium itself, follows from the varying velocity of different parts of the wave, as investigated at the beginning of this article.

It is easily shown that after a time we shall have to superpose on the original displacement a displacement proportional to the square of the particle velocity, and this will introduce just the same set of combination tones.

If w is the weight of a locomotive in tons, r the radius of curvature of the track, v the velocity in feet per sec.; then the horizontal force exerted on the bridge is wv 2 /gr tons.

Gounelle measured the velocity of electricity.

Let E be the effective elasticity of the aether; then E = pc t, where p is its density, and c the velocity of light which is 3 X 10 10 cm./sec. If = A cos" (t - x/c) is the linear vibration, the stress is E dE/dx; and the total energy, which is twice the kinetic energy Zp(d/dt) 2 dx, is 2pn2A2 per cm., which is thus equal to 1.8 ergs as above.

If we rest on the synthesis here described, the energy of the matter, even the thermal part, appears largely as potential energy of strain in the aether which interacts with the kinetic energy associated with disturbances involving finite velocity of matter.

But it is found not to vary at all, even up to the second order of the ratio of the earth's velocity to that of light.

We shall make the natural supposition that motion of the aether, say with velocity (u,v,w) at the point (x,y,z), is simply superposed on the velocity V of the optical undulations through that medium, the latter not being intrinsically altered.

If this relation is true along all paths, the velocity of the aether must be of irrotational type, like that of frictionless fluid.

As, however, our terrestrial optical apparatus is now all in motion along with the matter, we must dealt .with the rays relative to the moving system, and to these also Fermat's principle clearly applies; thus V+ (lu'--mv'-Fnw') is here the velocity of radiation in the direction of the ray, but relative to the moving material system.

This theory secures that the times of passage of the rays shall be independent of the motion of the system, only up to the first order of the ratio of its velocity to that of radiation.

Now the electric force (P,Q,R) is the force acting on the electrons of the medium moving with velocity v; consequently by Faraday's electrodynamic law (P,Q,R) = (P',Q' - vc, R'+vb) where (P',Q',R') is the force that would act on electrons at rest, and (a,b,c) is the magnetic induction.

If v varies with respect to locality, or if there is a velocity of convection (p,q,r) variable with respect to direction and position, and analytical expression of the relation (ii) assumes a more complex form; we thus derive the most general equations of electrodynamic propagation for matter treated as continuous, anyhow distributed and moving in any manner.

Trains of waves nearly but not quite homogeneous as regards wave-length will as usual be propagated as wave-groups travelling with the slightly different velocity d(VX-1)/dX-', the value of K occurring in V being a function of X determined by the law of optical dispersion of the medium.

According to these experiments, the resistance of the air can be represented by no simple algebraical law over a large range of velocity.

Abandoning therefore all a priori theoretical assumption, Bashforth set to work to measure experimentally the velocity of shot and the resistance of the air by means of equidistant electric screens furnished with vertical threads or wire, and by a chronograph which measured the instants of time at which the screens were cut by a shot flying nearly horizontally.

As a first result of experiment it was found that the resistance of similar shot was proportional, at the same velocity, to the surface or cross section, or square of the diameter.

The resistance R can thus be divided into two factors, one of which is d 2, where d denotes the diameter of the shot in inches, and the other factor is denoted by p, where p is the resistance in pounds at the same velocity to a similar I-in.

We first determine the time t in seconds required for the velocity of a shot, d inches in diameter and weighing w lb, to fall from any initial velocity V(f/s) to any final velocity v(f/s).

We put and call C the ballistic coefficient (driving power) of the shot, so that (6) At = COT, where (7) AT = Av/gp, and AT is the time in seconds for the velocity to drop Av of the standard shot for which C = I, and for which the ballistic table is calculated.

Since p is determined experimentally and tabulated as a function of v, the velocity is taken as the argument of the ballistic table; and taking Av =10, the average value of p in the interval is used to determine AT.

Denoting the value of T at any velocity v by T (v), then (8) T(v) = sum of all the preceding values of AT plus an arbitrary constant, expressed by the notation (9) T(v) =Z(Av)/gp+ a constant, or fdv/gp+ a constant, in which p is supposed known as a function of v.

The constant may be any arbitrary number, as in using the table the difference only is required of two tabular values for an initial velocity V and final velocity v; and thus (to) T(V) - T(v) = Ev Ov/gp or fvdv/gp; and for a shot whose ballistic coefficient is C (II) t=C[T(V) - T(v)].

Denoting by S(v) the sum of all the values of AS up to any assigned velocity v, (is) S(v) =E(OS)+ a constant, by which S(v) is calculated from AS, and then between two assigned velocities V and v, V AT, = vAv or rvvdv vgp gp' and if s feet is the advance of a shot whose ballistic coefficient is C, (17) s=C[S(V) - S(v)].

In an extended table of S, the value is interpolated for unit increment of velocity.

These functions, T, S, D, 1, A, are shown numerically in the following extract from an abridged ballistic table, in which the velocity is taken as the argument and proceeds by an increment of 10 f/s; the column for p is the one determined by experiment, and the remaining columns follow by calculation in the manner explained above.

In any region of velocity where it is possible to represent p with sufficient accuracy by an empirical formula composed of a single power of v, say v m, the integration can be effected which replaces the summation in (to), (16), and (24); and from an analysis of the Krupp experiments Colonel Zabudski found the most appropriate index m in a region of velocity as given in the following table, and the corresponding value of gp, denoted by f (v)or v m lk or its equivalent Cr, where r is the retardation.

Given the ballistic coefficient C, the initial velocity V, and a range of R yds.

The last column in the Range Table giving the inches of penetration into wrought iron is calculated from the remaining velocity by an empirical formula, as explained in the article Armour Plates.

Also the velocity v at the end of the arc is given by (87) ve = u e sec 0 cos n.

In this table (93) sin 20=Ca, where a is a function tabulated for the two arguments, V the initial velocity, and R/C the reduced range in yards.

This muzzle velocity is about 5% greater than the 2150 f/s of the range table, so on these considerations we may suppose about 10% of work is lost by friction in the bore; this is expressed by saying that the factor of effect is f =0.9.

According to the wave-theory of light, refraction is due to a change of velocity when light passes from one medium to another.

The phenomenon of dispersion shows that in dispersive media the velocity is different for lights of different wave-lengths.

In free space, light of all wave-lengths is propagated with the same velocity, as is shown by the fact that stars, when occulted by the moon or planets, preserve their white colour up to the last moment of disappearance, which would not be the case if one colour reached the eye later than another.

It can be shown mathematically that the velocity of propagation will be greatly increased if the frequency of the light-wave is slightly greater, and greatly diminished if it is slightly less than the natural frequency of the molecules; also that these effects become less and less marked as the difference in the two frequencies increases.

The great river receives an abundant load of silt from its tributaries, and takes up ano lays down silt from its own bed and banks with every change of velocity.

The load of silt borne down stream by the river finally, after many halts on the way, reaches the waters of the Gulf, where the decrease of velocity, aided by the salinity of the sea water, causes the formation of a remarkable delta, leaving less aggraded areas as shallow lakes (Lake Pontchartrain on the east, and Grand Lake on the west of the river).

At Canyon City it passes out of the Rockies through the Grand Canyon of the Arkansas; then turning eastward, and soon a turbid, shallow stream, depositing its mountain detritus, it flows with steadily lessening gradient and velocity in a broad, meandering bed across the prairies and lowlands of eastern Colorado, Kansas, Oklahoma and Arkansas, shifting its direction sharply to the south-east in central Kansas.

The great velocity of electrical transmission suggested the possibility of utilizing it for sending messages; and, after many experiments and the practical advice and business-like co-operation of William Fothergill Cooke (1806-1879), a patent for an electric telegraph was taken out in their joint names in 1837.

The tidal currents, or races, or roost (as some of them are called locally, from the Icelandic) off many of the isles run with enormous velocity, and whirlpools are of frequent occurrence, and strong enough at times to prove a source of danger to small craft.

Since the capacity of a stream to carry matter in suspension is proportional to its velocity, it follows that any circumstance tending to retard the rate of flow will induce deposition.

Thus a fall in the gradient at any point in the course of a stream; any snag, projection or dam, impeding the current; the reduced velocity caused by the overflowing of streams in flood and the dissipation of their energy where they enter a lake or the sea, are all contributing causes to alluviation, or the deposition of streamborne sediment.

Reinders (Ber., 1896, 29, p. 1369), who found that the reaction is monomolecular, and that the velocity constant of the reaction is proportional to the amount of the hydrochloride of the base present and also to the temperature, but is independent of the concentration of the diazoamine.

This apparent motion is due to the finite velocity of light, and the progressive motion of the observer with the earth, as it performs its yearly course about the sun.

If the bearer be stationary, rain-drops will traverse the tube without touching its sides; if, however, the person be walking, the tube must be inclined at an angle varying as his velocity in order that the rain may traverse the tube centrally.

Bradley recognized the fact that the experimental determination of the aberration constant gave the ratio of the velocities of light and of the earth; hence, if the velocity of the earth be known, the velocity of light is determined.

If N be the frequency of a homogeneous vibration sent out by a molecule at rest, the apparent frequency will be N (1 v/ V), where V is the velocity of light and v is the velocity of the line of sight, taken as positive if the distance from the observer increases.

If all molecules moved with the velocity of mean square, the line would be drawn out into a band having on the frequency scale a width 2Nv/V, where v is now the velocity of mean square.

If the motion were that of a body at white heat, or say a temperature of loco, the velocity of mean square would be 39co metres per second and the apparent width of the band would be doubled.

Hemsalech 1 have measured the velocity with which the luminous molecules are projected from metallic poles when a strong spark is passed through the air interval which separates the poles.

In the case of some metals, notably bismuth, the velocity measured was different for different lines, which seems intelligible only on the supposition that the metal vapour consists of different vibrating systems which can differ with different velocities.

The "mean moon" is a fictitious moon which moves around the earth with a uniform velocity and in the same time as the real moon.

In order to exert force, or at all events that force of reciprocal pressure which we best understand, and on which, in impact, the third law of motion was founded, there are always at least two bodies, enduring, triply extended, mobile, each inert, mutually impenetrable or resistent, different yet similar; and in order to have produced any effect but equilibrium, some bodies must at some time have differed either in mass or in velocity, otherwise forces would only have neutralized one another.

It shows that the bodies impress on one another opposite changes of velocity inversely as their weights or masses; and that in doing so they always begin by reducing one another to a joint mass with a common velocity, whatever they may do afterwards in consequence of their elasticities.

The two bodies therefore do not penetrate one another, but begin by acting on one another with a force precisely sufficient, instead of penetrating one another, to cause them to form a joint mass with a common velocity.

Bodies then are triply extended substances, each occupying enough space to prevent mutual penetration, and by this force of mutual impenetrability or interresistance cause one another to form a joint mass with a common velocity whenever they collide.

Withdraw this foundation of bodies as inter-resisting forces causing one another in collision to form a joint mass with a common velocity but without penetration, and the evidence of the third law disappears; for in the case of attractive forces we know nothing of their modus operandi except by the analogy of the collision of inter-resisting bodies, which makes us believe that something similar, we know not what, takes place in gravity, magnetism, electricity, &c. Now, Mach, though he occasionally drops hints that the discovery of the law of collision comes first, yet never explains the process of development from it to the third law of motion.

He has therefore lost sight of the truths that bodies are triply extended, mutually impenetrable substances, and by this force causes which reduce one another to a joint mass with a common velocity on collision, as for instance in the ballistic pendulum; that these forces are the ones we best understand; and that they are reciprocal causes of the common velocity of their joint mass, whatever happens afterwards.

The chief results we have found against idealism are that bodies have not been successfully analysed except into bodies, as real matter; and that bodies are known to exert reciprocal pressure in reducing one another to a joint mass with a common velocity by being mutually impenetrable, as real forces.

Galileo proceeded to measure the motion of a body on a smooth, fixed, inclined plane, and found that the law of constant acceleration along the line of slope of the plane still held, the acceleration decreasing in magnitude as the angle of inclination was reduced; and he inferred that a body, moving on a smooth horizontal plane, would move with uniform velocity in a straight line if the resistance of the air, and friction due to contact with the plane, could be eliminated.

Such statements as that a body moves in a straight line, and that it has a certain velocity, have no meaning unless the base, relative to which the motion is to be reckoned, is defined.

Newton assumed the possibility of choosing a base such that, relatively to it, the motion of any particle would have only such divergence from uniform velocity in a straight line as could be expressed by laws of acceleration dependent on its relation to other bodies.

Suppose two small smooth spherical bodies which can be regarded as particles to be brought into collision, so that the velocity of each, relative to any base which is unaffected by the collision, is suddenly changed.

The additions of velocity which the two bodies receive respectively, relative to such a base, are in opposite directions, and if the bodies are alike their magnitudes are equal.

If the bodies though of the same substance are of different sizes, the magnitudes of the additions of velocity are found to be inversely proportional to the volumes of the bodies.

In fact, experiments upon the changes of velocity of bodies, due to a mutual influence between them, bring to light a property of bodies which may be specified by a quantity proportional to their volumes in the case of bodies which are perceived by other tests to be of one homogeneous substance, but otherwise involving also another factor.

When, as in the case of contact, a mutual relation is perceived between the motions of two particles, the changes of velocity are in opposite directions, and the ratio of their magnitudes determines the ratio of the masses of the particles; the motion being reckoned relative to any base which is unaffected by the change.

This test involves only changes of velocity, and so does not distinguish between two bases, each of which moves relatively to the other with uniform velocity without rotation.

Hence the force required to drive one gramme-molecule of sugar through water with a velocity of one centimetre per second may be calculated as some thousands of millions of kilogrammes weight.

The resistance offered by the liquid, and therefore the force F, required to drive one grammemolecule through the liquid with unit velocity is the sum of the corresponding quantities for the individual ions.

This is precisely the number found from the velocity of sound in argon as determined by Kundt's method, and it leaves no room for any sensible energy of rotatory or vibrational motion.

The electromotive force is practically constant no matter what the velocity of the disks, but according to some observers the internal resistance decreases as the velocity increases.

The mean velocity of their flow seldom exceeds 4.9 ft., but rises to 6.4 ft.

In the lower reaches of the streams the velocity and slope are of course affected by the tides.

The reason why the frictional resistance would be further increased is the very simple one that the increase in the rate of production implies directly a corresponding increase in the quantity of blast forced through, and hence in the velocity of the rising gases, because the chemical work of the blast furnace needs a certain quantity of blast for each ton of iron made.

In short, to increase the rate of production by lengthening the furnace increases the frictional resistance of the rising gases, both by increasing their quantity and hence their velocity and by lengthening their path.

We see how powerful must be the lifting effect of the rising gases when we reflect that their velocity in a too ft.

Conceive these gases passing at this great velocity through the narrow openings between the adjoining lumps of coke and ore.

Indeed, the velocity must be far greater than this where the edge or corner of one lump touches the side of another, and the only room for the passage of this enormous quantity of gas is that left by the roughness and irregularity of the individual lumps.

If the linear velocity of the cups in feet a second is V 1, and the linear velocity of the jet is V2, then the velocity of the jet relative to the cup is V2 - V1 feet a second, and if the whole energy of the water is to be given up to the cups, the water must leave the cup with zero absolute velocity.

But its velocity relative to the cup, as it passes backwards, is - (V 2 - V 1), and since the forward velocity of the cup is Vi, the absolute velocity of the water is - (V2 - Vi) +VI or2V i - V2.

This will become zero if V 1 is 2V 2, that is, if the linear velocity of the cupcentres is one-half that of the jet of water impinging upon them.

The tides, which are very high, run into it with amazing velocity, but at low water the bottom is left nearly dry for some distance below the latitude of the town of Cambay.

The velocity of propagation of temperature waves will be the same under similar conditions in two substances which possess the same diffusivity, although they may differ in conductivity.

Uniformity of temperature could only be secured by using a high velocity of flow, or violent stirring.

For instance, the velocity of propagation of a wave having a period of a day is nearly twenty times as great as that of a wave with a period of one year; but on the other hand the penetration of the diurnal wave is nearly twenty times less, and the shorter waves die out more rapidly.

The magnitude of the stress per unit area parallel to the direction of flow is evidently proportional to the velocity gradient, or the rate of change of velocity per cm.

If the effects depended merely on the velocity of translation of the molecules, both conductivity and viscosity should increase directly as the square root of the absolute temperature; but the mean free path also varies in a manner which cannot be predicted by theory and which appears to be different for different gases (Rayleigh, Proc. R.S., January 1896).

An auroral curtain travelling with considerable velocity would approach from the south, pass right overhead and retire to the north.

According to numerous observations made at Cape Thorsden, the apparent angular velocity of arcs increases on the average with their altitude.

The velocity 109.09 was much the largest observed, the next being 52.38; both were from observations lasting under half a minute.

There is also difficulty in ensuring that the observations shall be simultaneous, an important matter especially when the apparent velocity is considerable.

Cathode rays usually have a velocity about a tenth that of light, but in exceptional cases it may approach a third of that of light.

Hertzian waves have the velocity of light itself.

Now if a be the amplitude expressed in millimetres, and t the period expressed in seconds, then the maximum velocity of an earth particle as it vibrates to and fro equals 27ra/t, whilst the maximum acceleration equals 4,r 2 0 2.

For example, if a body, say a coping-stone, has been thrown horizontally through a distance a, and fallen from a height b, the maximum horizontal velocity with which it was projected equals !

Another remarkable phenomenon is the zobaa, a lofty whirlwind of sand resembling a pillar, which moves with great velocity.

The mean velocity of winds for 1906 was 110 m.; the maximum recorded being 148 in May, the minimum velocity recorded being 76 in December.

The optical apparatus generally consists of a mirror mounted on an axis parallel to the axis of the earth, and rotated with the same angular velocity as the sun.

It is easily seen that if the mirror be rotated at the same angular velocity as the sun the right ascensions will remain equal throughout the day, and therefore this device reflects the rays in the direction of the earth's axis; a second fixed mirror reflects them in any other fixed direction.

By adjusting the right ascension of the plane ABC and rotating the axis with the angular velocity of the sun, it follows that BC will be the direction of the solar rays throughout the day.

By the use of a revolving mirror similar to that used by Sir Charles Wheatstone for measuring the rapidity of electric currents, he was enabled in 1850 to demonstrate the greater velocity of light in air than in water, and to establish that the velocity of light in different media is inversely as the refractive indices of the media.

With Wheatstone's revolving mirror he in 1862 determined the absolute velocity of light to be 298,000 kilometres (about 185,000 m.) a second, or 10,000 kilom.

Further, by causing the hour circle, and with it the polar axis, to rotate by clockwork or some equivalent mechanical contrivance, at the same angular velocity as the earth on its axis, but in the opposite direction, the telescope will, apart from the effects of refraction, automatically follow a star from rising to setting.

The Direct Methods Of Measuring The Ratio S/S, By The Velocity Of Sound And By Adiabatic Expansion, Are Sufficiently Described In Many Text Books.

The suggestion was made, and seems to be the true explanation, that what was actually witnessed was the wave of light due to the outburst of the nova, spreading outwards with its velocity of 186,000 m.

By means of the spectroscope it is possible to determine the relative orbital velocity of the two components, and this when compared with the period fixes the absolute dimensions of the orbit; the apparent dimensions of the orbit being known from visual observations the distance can then be found.

The velocity in the line of sight can be determined by spectroscopic observation, so that in a few cases the motion of the star is completely known.

Probably the velocity of Arcturus is also over 100 m.

Campbell the average velocity in space of a star is 21.2 m.

Regarded as a linear velocity, the parallactic motion is the same for all stars, being exactly equal and opposite to the solar motion; but its amount, as measured by the corresponding angular displacement of the star, is inversely proportional to the distance of the star from the earth, and foreshortening causes it to vary as the sine of the angular distance from the apex.

Campbell from the radial motions of 280 stars found the velocity to be 20 kilometres per second with a probable error of 12 km.

Halm deduced a velocity of 20.8 km.

In 1728 was published "A Letter from Dr Clarke to Benjamin Hoadly, F.R.S., occasioned by the controversy relating to the Proportion of Velocity and Force in Bodies in Motion," printed in the Philosophical Transactions.

At Vienna he had lessons in pianoforte playing from Carl Czerny of " Velocity " fame, and from Salieri in harmony and analysis of scores.

This is called the curve of positions or space-time curve; its gradient represents the velocity.

It is a matter of ordinary observation that different bodies acted on by the same force, or what is judged to be the same force, undergo different changes of velocity in equal times.

The product mu of the mass into the velocity is called the momentum or (in Newtons phrase) the quantity of motion.

On the Newtonian system the motion of a particle entirely uninfluenced by other bodies, when referred to a suitable base, would be rectilinear, with constant velocity.

If we take as rough values a=21 X,o6 feet, g=32 foot-second units, we get a velocity of 36,500 feet, or about seven miles, per second.

We may briefly notice the case of resistance varying as the square of the velocity, which is mathematically simple.

Tait that a similar representation of the type (30) is obtained if we replace the circle by an equiangular spiral described, with a constant angular velocity about the pole, in the direction of diminishing radius vector.

For purposes of mathematical treatment a force which produces a finite change of velocity in a time too short to be appreciated is regarded as infinitely great, and the time of action as infinitely short.

Thus the unit of velocity is that of a point describing the unit of length in the unit of time; it may be denoted by LTi, this symbol indicating that the magnitude of the unit in question varies directly as the unit of length and inversely as the unit of time.

The unit of acceleration is the acceleration of a point which gains unit velocity in unit time; it is accordingly denoted by LT2.

As 6t is indefinitely diminished, the vector OU will tend to a definite limit OV; this is adopted as the definitiov of the velocity of the moving point at the instant t.

The momentum of a particle is the vector obtained by multiplying the velocity by the mass in.

In symbols, if v be the velocity and p the perpendicular from 0 to the tangent to the path, pv=h, (1)

Hence the character of the orbit depends on whether the velocity at any point is, less than, equal to, or greater than the velocity from infinity, as it is called.

In order that the spiral may be described it is necessary that the velocity of projection should be adjusted to make h=iju.

If A or B vanish we have an equiangular spiral, and the velocity at infinity is zero.

A point on a central orbit where the radial velocity (drfdt) vanishes is called an apse, and the corresponding radius is called an apse-line.

If the force is always the same at the same distance any apse-line will divide the orbit symmetrically, as is seen by imagining the velocity at the apse to be reversed.

If in a central orbit the velocity is equal to the velocity from infinity, we have, from (5),

The question presents itself whether ther then is any other law of force, giving a finite velocity from infinity, under which all finite orbits are necessarily closed curves.

At the beginning of 13 the velocity of a moving point P was represented by a vector OV drawn from a fixed origin 0.

The locus of the point V is called the hodograp/z (q.v.); and it appears that the velocity of the point V along the hodograph represents in magnitude and in directon tbt acceleration in the original orbit.

In the motion of a projectile under gravity the hodograph is a vertical line described with constant velocity.

In elliptic harmonic motion the velocity of P is parallel and proportional to the semi-diameter CD which is conjugate to the radius CP; the hodograph is therefore an ellipse similar to the actual orbit.

In the case of a particle oscillating under gravity on a smooth cycloid from rest at the cusp the hotlograph is a circle through the pole, described with constant velocity.

For example, the mass-centre of I system free from extraneous force will describe a straight lin with constant velocity.

Then dU/dt, =w say, is the angular velocity of the body.

The angular velocity being constant, the effective force on a particle m at a distance r from Oz is snw2r toward& this axis, and its components are accordingly w2mx, wfmy, 0.

If the extraneous forces have zero moment about G the angular velocity 0 is constant.

The circle is described with the constant angular velocity o.

As an example of this latter type, suppose that a sphere is placed on the highest point of a fixed sphere and set spinning about the vertical diameter with the angular velocity n; it will appear that under a certain condition the motion of G consequent on a slight disturbance will be oscillatory.

Now T = 3/41w1, where w is the angular velocity and I is the moment of inertia about the instantaneous axis.

The motion of the body relative to 0 is therefore completely represented if we imagine the momental ellipsoid at 0 to roll without sliding on a plane fixed in space, with an angular velocity proportional at each instant to the radius-vector of the point of contact.

The angular velocity (r) about this axis is then constant.

As a first application of the equations (2) take the case of a solid constrained to rotate with constant angular velocity to about a fixed axis (1, m, n).

The physical characteristics of a normal mode are that an impulse of a particular normal type generates an initial velocity of that type only, and that a constant extraneous force of a particular normal type maintains a displacement of that type only.

It consists of two elements, the velocity ratio, which is the ratio of any two magnitudes bearing to each other the proportions of the respective velocities of the two points at a given instant, and the directional relation, which is the relation borne to each other by the respective directions of the motions of the two points at the same given instant.

The comparative motion of two points at a given instant is capable of being completely expressed by one of Sir William Hamiltons Quaternions,the tensor expressing the velocity ratio, and the versor the directional relation.

Let a represent the area of the section of a piston made by a plane perpendicular to its direction of motion, and v its velocity, which is to be considered as positive when outward, and negative when inward.

Let the angular velocity of the rotation be denoted by a=dO/dt, then the linear velocity of any point A at the distance r from the axis is or; and the path of that point is a circle of the radius r described about the axis.

This is the principle of the modification of motion by the lever, which consists of a rigid body turning about a fixed axis called a fulcrum, and having two points at the same or different distances from that axis, and in the same or different directions, one of which receives motion and the other transmits motion, modified in direction and velocity according to the above law.

Velocity Ratio of Components of Motion.As the distance between any two points in a rigid body is invariable, the projections of their velocities upon the line joining them must be equal.

The line T on the surface bbb has for the instant no velocity it a direction perpendicular to AB; becau2e for the instant it touches, without sliding, the line T on the fixed surface aaa.