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166. It may be shewn in a similar manner that the time of describing about the focus an arc of an ellipse or hyperbola whose chord and extreme radii vectores are given, may be expressed in terms of these quantities and the axis major alone. For the proof we must refer to the Mécanique Céleste, or to Pontécoulant's Système du Monde.

It may also be shewn, in much the same manner, that the ratio of the area described in a given time, to that of the triangle formed by the chord and extreme radii vectores, may be expressed independently of the parameter of the path.

EXAMPLES.

(1) If the perihelion distance of a comet's orbit be of the radius of the Earth's orbit supposed circular, find the number of days the comet will remain within the Earth's orbit.

(2) If a comet describe 90° from perihelion in 100 days, compare its perihelion distance with the distance of a planet which describes its circular orbit in 942 days.

(3) Shew how to divide a planet's elliptic orbit by a diameter, so that the times of describing the two parts are as n: 1, and find in what cases only one such line can be drawn.

(4) In the case of planets and comets prove the following formulæ, the letters being the same as in the text,

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- 2 (λ cos u + cos 2u + cos 3u+ &c.)

(5) A body describes an ellipse: prove that the times of describing the two parts, into which the orbit is divided by the axis minor, are to one another as π + 2e is to π — 2e, where e is the excentricity of the ellipse.

(6) If Pp, Qq be chords parallel to the axis major of an elliptic orbit, shew that the difference of the times through the arcs PQ, pq varies as the distance between the chords.

(7) If a comet whose orbit is inclined to the plane of the ecliptic were observed to pass over the Sun's disc, and three months after to strike the planet Mars, determine its distance from the Earth at the first observation, the Earth and Mars describing about the Sun circles in the same plane whose radii are as 2 : 3.

(8) Shew that the arithmetic mean of the distances of a planet from the Sun, at equal indefinitely small intervals of time, is

a (1 + 1).

(9) When a body describes an ellipse under the action of a force in the focus S, if H be the other focus, the square

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(10) The time through an arc of a parabolic orbit bounded by a focal chord ∞ (chord)3.

(11) If a circle be described passing through the focus and vertex of a parabolic orbit, and also through the position of the moving particle at each instant, shew that its center describes with uniform velocity a straight line bisecting at right angles the perihelion distance.

(12) Shew that the velocity of a comet perpendicular to the major axis varies inversely as its radius vector.

2

(13) D, D, being two distances of a comet, on opposite sides of perihelion, including a known angle, shew that the position of perihelion may be found from the equation

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= tan(sum of true anomalies). tan (difference).

(14) In what point of all conic sections is the paracentric velocity a maximum? Shew that in such a case the velocity is to that in a circle at the same distance as the distance is to the perpendicular on the tangent.

(15) In an elliptic orbit find the relation between the mean angular velocity about the center of force and the angular velocity about the other focus, and thence shew that when e is small the latter is nearly uniform.

(16) If a, ẞ be the greatest and least angular velocities in an ellipse about the focus, the mean angular velocity is

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(17) Find the maximum value of 0-nt in an elliptic orbit, and develop it in powers of e, shewing that it cannot contain even powers.

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CHAPTER VII.

CONSTRAINED MOTION.

167. WE come now to the case of the motion of a particle subject to the action not only of given forces, but of undetermined pressures or tensions. Such cases occur when the particle is attached to a fixed, or moving, point by means of a rod or string, and when it is forced to move on a curve or surface.

In applying to a problem of this kind the general equations of motion of a free particle, we must assume directions and intensities for the unknown forces, treating them then as known, and it will always be found that the geometrical circumstances of the motion will furnish the requisite number of additional equations for the determination of all the unknown quantities in terms of the time.

One case of this kind has been already treated of (§ 79), namely, that of a particle moving on an inclined plane under the action of gravity. There the undetermined force is the pressure on the plane, which however is evidently constant, and equal to the resolved part of the particle's weight perpendicular to the plane.

The laws of kinetic friction are but imperfectly known, and the few investigations which will be given of motion on a rough curve or surface are of very slight importance.

168. The simplest case is

A particle is constrained to move on a given smooth plane curve, under the action of given forces in the plane of the curve, to determine the motion.

Taking rectangular axes in this plane, the forces may be resolved into two, X, Y, parallel respectively to the axes of and y. In addition there will be the force R, the mutual pressure between the curve and particle, which evidently acts in the normal to the curve since there is no friction.

Let P be the position of the particle at the time t; and let

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the forces X, Y, R, act as in the figure. Draw PT, a tangent to the constraining curve at P. Then if PTx = 0, we have

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The mass of the particle being, as before, taken as unity, the equations of motion are

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These two equations, together with the equation to the given curve, are sufficient to determine the motion completely.

To eliminate R, multiply (1) by

dx

dt

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We thus obtain,

(2) by
dt

dy

and add.

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(since dy de

dx

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ds

dt'

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