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thus the point moves in the plane. For the proof of the second part of the proposition we must make a slight digression.

46. The Moment of a velocity or of a force about any point is the product of its magnitude into the perpendicular from the point upon its direction. The moment of the resultant velocity of a par ticle about any point in the plane of the components is equal to the algebraic sum of the moments of the components, the proper sign of each moment depending on the direction of motion about the point. The same is true of moments of forces and of moments of momentum, as defined in Chapter II.

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First, consider two component motions, AB and AC, and let AD be their resultant (§ 31). Their half-moments round the point are respectively the areas OAB, OCA. Now OCA, together with half the area of the parallelogram CABD, is equal to OBD. Hence the sum of the two half-moments together with half the area of the parallelogram is equal to AOB together with BOD, that is to say; to the area of the whole figure OABD. But ABD, a part of this figure, is equal to half the area of the parallelogram; and therefore the remainder, OAD, is equal to the sum of the two half-moments. But OAD is half the moment of the resultant velocity round the point 0. Hence the moment of the resultant is equal to the sum of the moments of the two components. By attending to the signs of the moments, we see that the proposition holds when O is within the angle CAB.

B

If there be any number of component rectilineal motions, we may compound them in order, any two taken together first, then a third, and so on; and it follows that the sum of their moments is equal to the moment of their resultant. It follows, of course, that the sum of the moments of any number of component velocities, all in one plane, into which the velocity of any point may be resolved, is equal to the moment of their resultant, round any point in their plane. It follows also, that if velocities, in different directions all in one plane, be successively given to a moving point, so that at any time its velocity is their resultant, the moment of its velocity at any time is the sum of the moments of all the velocities which have been successively given to it. 47. Thus if one of the components always passes through the point, its moment vanishes. This is the case of a motion in which the acceleration is directed to a fixed point, and we thus prove the second theorem of § 45, that in the case supposed the areas described by the radius-vector are proportional to the times; for, as we have seen, the moment of the velocity is double the area traced out by the radius-vector in unit of time.

48. Hence in this case the velocity at any point is inversely as the perpendicular from the fixed point to the tangent to the path or the momentary direction of motion.

For the product of this perpendicular and the velocity at any instant gives double the area described in one second about the fixed point, which has just been shown to be a constant quantity.

Other examples of these principles will be met with in the chapters on Kinetics.

49. If, as in § 35, from any fixed point, lines be drawn at every instant representing in magnitude and direction the velocity of a point describing any path in any manner, the extremities of these lines form a curve which is called the Hodograph. The fixed point from which these lines are drawn is called the hodographic origin. The invention of this construction is due to Sir W. R. Hamilton; and one of the most beautiful of the many remarkable theorems to which it leads is this: The Hodograph for the motion of a planet or comet is always a circle, whatever be the form and dimensions of the orbit. The proof will be given immediately.

It was shown (§ 35) that an arc of the hodograph represents the change of velocity of the moving point during the corresponding time; and also that the tangent to the hodograph is parallel to the direction, and the velocity in the hodograph is equal to the amount of the acceleration of the moving point.

When the hodograph and its origin, and the velocity along it, or the time corresponding to each point of it, are given, the orbit may easily be shown to be determinate.

[An important improvement in nautical charts has been suggested by Archibald Smith1. It consists in drawing a curve, which may be called the tidal hodograph with reference to any point of a chart for which the tidal currents are to be specified throughout the chief tidal period (twelve lunar hours). Numbers from I. to XII. are placed at marked points along the curve, corresponding to the lunar hours. Smith's curve is precisely the Hamiltonian hodograph for an imaginary particle moving at each instant with the same velocity and the same direction as the particle of fluid passing, at the same instant, through the point referred to.]

50. In the case of a projectile (§ 44), the horizontal velocity is unchanged, and the vertical velocity increases uniformly. Hence the hodograph is a vertical straight line, whose distance from the origin is the horizontal velocity, and which is described uniformly. 51. To prove Hamilton's proposition (§ 49), let APB be a portion of a conic section and S one focus. Let P move so that SP describes equal areas in equal times, that is (§ 48), let the velocity be inversely as the perpendicular SY from S to the tangent to the orbit. If ABP be an ellipse or hyperbola, the intersection of the perpendicular with the tangent lies in the circle YAZ, whose diameter is the major axis. Produce YS to cut the circle again in Z. Then YS SZ is constant, and therefore SZ is inversely as SF, that is, SZ is proportional to the velocity at P. Also 1 Proc. R. S. 1865.

A

S

P

U

-B

SZ is perpendicular to the direction of motion PY, and thus the circular locus of Z is the hodograph turned through a right angle about S in the plane of the orbit. If APB be a parabola, AY is a straight line. But if another point U be taken in YS produced, so that YS SU is constant, the locus of U is easily seen to be a circle. Hence the proposition is generally true for all conic sections. The hodograph surrounds its origin if the orbit be an ellipse, passes through it when the orbit is a parabola, and the origin is without the hodograph if the orbit is a hyperbola.

52. A reversal of the demonstration of § 51 shows that, if the acceleration be towards a fixed point, and if the hodograph be a circle, the orbit must be a conic section of which the fixed point is a focus.

But we may also prove this important proposition as follows: Let A be the centre of the circle, and O the hodographic origin. Join OA and draw the perpendiculars

PM to OA and ON to PA. Then OP
is the velocity in the orbit: and ON, being
parallel to the tangent at P, is the direc-
tion of acceleration in the orbit; and is
therefore parallel to the radius-vector to
the fixed point about which there is equable
The velocity parallel
description of areas.
to the radius-vector is therefore ON, and
the velocity perpendicular to the fixed line.

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N

A

M

Hence, in the orbit, the velocity along the radius-vector is proportional to that perpendicular to a fixed line: and therefore the radius-vector of any point is proportional to the distance of that point from a fixed line-a property belonging exclusively to the conic sections referred to their focus and directrix.

53. The path which, in consequence of Aberration, a fixed star seems to describe, is the hodograph of the earth's orbit, and is therefore a circle whose plane is parallel to the plane of the ecliptic.

54. When a point moves in any manner, the line joining it with a fixed point generally changes its direction. If, for simplicity, we consider the motion to be confined to a plane passing through the fixed point, the angle which the joining line makes with a fixed line in the plane is continually altering, and its rate of alteration at any instant is called the Angular Velocity of the first point about the second. If uniform, it is of course measured by the angle described in unit of time; if variable, by the angle which would have been described in unit of time if the angular velocity at the instant in question were maintained constant for so long. In this respect the

process is precisely similar to that which we have already explained for the measurement of velocity and acceleration.

We may also speak of the angular velocity of a moving plane with respect to a fixed one, as the rate of increase of the angle contained by them; but unless their line of intersection remain fixed, or at all events parallel to itself, a somewhat more laboured statement is required to give a complete specification of the motion.

55. The unit angular velocity is that of a point which describes, or would describe, unit angle about a fixed point in unit of time. The usual unit angle is (as explained in treatises on plane trigonometry) that which subtends at the centre of a circle an arc whose length is equal 180°

to the radius; being an angle of nearly.

П

=57°29578...=57°17′44′′.8

56. The angular velocity of a point in a plane is evidently to be found by dividing the velocity perpendicular to the radius-vector by the length of the radius-vector.

57. When the angular velocity is variable its rate of increase or diminution is called the Angular Acceleration, and is measured with reference to the same unit angle.

58. When one point describes uniformly a circle about another, the time of describing a complete circumference being T, we have the angle 27 described uniformly in T; and, therefore, the angular velocity is Even when the angular velocity is not uniform, as in 2π T


T'

a planet's motion, it is useful to introduce the quantity which is then called the mean angular velocity.

59. When a point moves uniformly in a straight line its angular velocity evidently diminishes as it recedes from the point about which the angles are measured, and it may easily be shown that it varies inversely as the square of the distance from this point. The same proposition is true for any path, when the acceleration is towards the point about which the angles are measured: being merely a different mode of stating the result of § 48.

60. The intensity of heat and light emanating from a point, or from a uniformly radiating spherical surface, diminishes according to the inverse square of the distance from the centre. Hence the rate at which a planet receives heat and light from the sun varies in simple proportion to the angular velocity of the radius-vector. Hence the whole heat and light received by the planet in any time is proportional to the whole angle turned through by its radius-vector in the same time.

61. A further instance of this use of the idea of angular velocity may now be given, to solve the problem of finding the hodograph (§ 35) for any case of motion in which the acceleration is directed to a fixed point, and varies inversely as the square of the distance from that point. The velocity of P, in the hodograph PQ, being the

acceleration in the orbit, varies inversely as the square of the radius-vector; and therefore (§ 59) directly as the angular velocity. Hence the arc of PQ, described in any time, is proportional to the corresponding angle-vector in the orbit, i.e. to the angle through which the tangent to PQ has turned. Hence (§ 9) the curvature of PQ is constant, or PQ is a circle.

This demonstration, reversed, proves that if the hodograph be a circle, and the acceleration be towards a fixed point, the acceleration varies inversely as the square of the distance of the moving point from the fixed point.

62. From §§ 61, 52, it follows that when a particle moves with acceleration towards a fixed point, varying inversely as the square of the distance, its orbit is a conic section, with this point for one focus, And conversely (§§ 47, 51, 62), if the orbit be a conic section, the acceleration, if towards either focus, varies inversely as the square of the distance: or, if a point moves in a conic section, describing equal areas in equal times by a radius-vector through a focus, the acceleration is always towards this focus, and varies inversely as the square of the distance.

63. All motion that we are, or can be, acquainted with, is Relative merely. We can calculate from astronomical data for any instant the direction in which, and the velocity with which, we are moving on account of the earth's diurnal rotation. We may compound this with the (equally calculable) velocity of the earth in its orbit. This resultant again we may compound with the (roughly-known) velocity of the sun relatively to the so-called fixed stars; but, even if all these elements were accurately known, it could not be said that we had attained any idea of an absolute velocity; for it is only the sun's relative motion among the stars that we can observe; and, in all probability, sun and stars are moving on (it may be with inconceivable rapidity) relatively to other bodies in space. We must therefore consider how, from the actual motions of a set of bodies, we may find their relative motions with regard to any one of them; and how, having given the relative motions of all but one with regard to the latter, and the actual motion of the latter, we may find the actual motions of all. The question is very easily answered. Consider for a moment a number of passengers walking on the deck of a steamer. Their relative motions with regard to the deck are what we immediately observe, but if we compound with these the velocity of the steamer itself we get evidently their actual motion relatively to the earth. Again, in order to get the relative motion of all with regard to the deck, we eliminate the motion of the steamer altogether; that is, we alter the velocity of each relatively to the earth by compounding with it the actual velocity of the vessel taken in a reversed direction.

Hence to find the relative motions of any set of bodies with regard

C

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