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velocity at any point is inversely proportional to the distance from the center of force; shew that its path will be a logarithmic spiral.

(29) A particle is describing a curve about a center of

1

force, and its velocity, find the law of force and the equation to the path.

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(30) A particle is projected in any direction from one extremity of a uniform straight line each particle of which attracts it with a force proportional to the distance, prove that the particle will pass through the other extremity.

(31) A particle projected in a given direction with a given velocity and attracted towards a given center of force has its velocity at every point to the velocity in a circle at the same distance as 1 to 2; find the orbit described, the position of the apse, and the law of force.

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(32) A particle is projected from a given point with a given velocity and is acted on by a central force varying inversely as the square of the distance; shew that whatever be the direction of projection the center of the orbit described will lie on the surface of a certain sphere.

(33) Find the locus of the center of force that a cycloid may be described with uniform velocity, and find the law of force to the moving center.

(34) If a particle revolve in a circle of radius r, about a center of force distant a from the center of the circle, shew that the time from distance r to the nearer apse is

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where & is the initial force; and that the periodic time is

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(35) Shew that the only law of central force for which the velocity at each point of the orbit can be equal to that in a circle at the same distance is that of the inverse third power, and that the orbit is the logarithmic spiral.

(36) A particle describes an equilateral hyperbola about a center of force in the center, shew that an angle from the apsidal line is connected with the time t of its description by

the formula

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(37) If a number of particles, describing different circles in the same plane about a center of force oc D3, start together from the same radius, find the curve in which they all lie when that which moves in the circle whose radius is a has completed a revolution.

(38) If the mth power of the periodic time be proportional to the nth power of the velocity in a circle, find the law of force in terms of the radius.

(39) If v be the velocity of a particle revolving in an ellipse about the center, v' its velocity when the direction of its motion is at right angles to the former direction, the time of describing the intercepted arc =

1
sin-
μ

vo' pab

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1

(40) A particle revolves in a circle about a center of force in the center, the force; the absolute force is suddenly increased in the ratio of m: 1 when the particle is at any assigned point of its path, and when the particle arrives again at the same point the absolute force is again increased in the same ratio; shew that the path which the particle will describe is an ellipse whose excentricity

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(41) In a curve described by a particle under the action of a central force the angle between the radius vector and the tangent varies as the time. Find the curve and law of force.

(42) Shew that the apsidal angle is the same for different apsidal distances, only when the force is as some power of the distance.

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that in the particular case of the projection being made at

h

α

distance a, and with velocity =√2, the equation to the orbit is

ra (1+0).

(44) Force, and a particle is projected from an apse at distance a with velocity, shew that the path is a cardioide, and that the periodic time is

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(45) A particle is revolving in an ellipse about a center of force in the focus; supposing that every time the particle

138

arrives at the lower apse the absolute force is diminished in the ratio of 1 to 1-n; find the excentricity of the elliptic orbit after p revolutions, the original excentricity being e.

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(46) A particle describes a circular orbit about a center of force situated in the center of the circle; prove that the form of the orbit will be stable or unstable according as the for ua, is less or not less than 3, P being the central force, u the reciprocal of the radius vector, and

value of
d log P
d log u

1

a

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the radius of the circle.

(47) If the equation for determining the apsidal distances in a central orbit contain the factor (u - a)", shew that u = a cannot correspond to an apse unless p be of one of the forms 4m + 2 2n + 1

4m+2 or

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If the factor u

will be a root of the equation

¢ (u) − h*i = 0,

where (u) is the central force.

a occur twice, then a

Shew

(48) Examine carefully the case of an apse where the center of force coincides with the center of curvature. that the particle will, after passing such an apse, describe a circle about the center of force, but that the motion will be unstable.

CHAPTER VI.

ELLIPTIC MOTION.

150. In this chapter we propose to deduce from the results of the last some of the properties of Elliptic and Parabolic Orbits described about a center of force in the focus. This is a problem of great interest, as it has been proved by actual observation that the orbits of planets and comets are in general (neglecting the small effects of disturbing forces) ellipses either very slightly excentric, or so much so as to be scarcely distinguishable from parabolas. There are, it is true, some comets whose orbits are moderately excentric ellipses, and some whose orbits are hyperbolas; but, as the problem in their case becomes very complicated, and the approximate methods which we will here employ are inapplicable to their motions, it has been considered advisable to omit the consideration of such cases.

151. For the intelligibility of what follows it will be necessary to premise a few definitions.

Suppose APA' to be an elliptic orbit described about a center of force in the focus S. Also suppose P to be the position of the particle at any time t. Draw PM perpendicular to the major axis AČA', and produce it to cut the auxiliary circle in the point Q. Let C be the common center of the curves. Join CQ.

When the moving particle is at A, the nearest point of the orbit to S, it is said to be in Perihelion.

The angle ASP, or the excess of the particle's longitude over that of the perihelion, is called the True Anomaly. Let us denote it by 0.

The angle ACQ is generally denoted by u.

called the Excentric Anomaly, and is And if

be the time of a complete

n

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