We have assumed that the earth is a sphere, and that the attraction which it exerts on a body placed at any point on the surface is directed towards the centre; but these assumptions are not strictly accurate, so that the result must not be considered absolutely true.

EXAMPLES. XIV.

1. Find the force towards the centre required to make a body move uniformly in a circle whose radius is 5 feet, with such a velocity as to complete a revolution in 5 seconds.

2. A stone of one lb. weight is whirled round horizontally by a string two yards long having one end fixed: find the time of revolution when the tension of the string is 3 lbs.

3. A body weighing P lbs. is at one end of a string, and a body weighing lbs. at the other; the system is in motion on a smooth horizontal table, P and Q describing circles with uniform velocities: determine the position of the point in the string which does not move.

4. A string 7 feet long can just support a weight of P lbs. without breaking; one end of the string is fixed to a point on a smooth horizontal table; a weight of lbs. is fastened to the other end and describes a circle with uniform velocity round the fixed point as centre: determine the greatest velocity which can be given to the weight of Qlbs. so as not to break the string.

XV. Motion in a conic Section round a focus.

173. The cases of motion which we shall discuss in the present Chapter are of great interest on account of the application of them to the earth and planets which describe ellipses round the sun in a focus.

In the remainder of this work we shall consider the action of a force on a given body, so that we shall be occupied only with the influence of the force on the velocity of the body: see Arts. 14, 45.

174. If a body describes an ellipse under the action of a force in a focus, the velocity at any point can be resolved into two, both constant in magnitude, one perpendicular to the major axis of the ellipse, and the other at right angles to the radius drawn from the body to the focus.

Let S be the focus which is the centre of force, H the other focus, P any point on the ellipse, SY and HZ perpendiculars from S and H on the tangent at P. Let C be the centre of the ellipse, A one end of the axis major.

A

Y

Р

Ꮓ

Н

By Art. 158 the velocity at P varies inversely as SY, and therefore directly as HZ; for SY× HZ is constant, by a property of the ellipse. Thus HZ may be taken to represent the velocity in magnitude, and it is at right angles to the velocity in direction. Now a velocity represented by HZ may be resolved into two represented by HC and CZ. And by the nature of the ellipse CZ is parallel to SP and equal to CA.

Hence a velocity represented by HZ in magnitude, and at right angles to HZ in direction, may be resolved into two, one represented by CA in magnitude and at right

angles to SP in direction, and the other represented by HC in magnitude and perpendicular to HS in direction.

It is convenient to have expressions for the magnitudes of these component velocities. Let CA-a, let b denote half the minor axis, and let e be the excentricity of the ellipse. Let h represent twice the area described by the radius SP in a unit of time; then the velocity at h.HZ h.HZ. by the nature of the ellipse. There

P=

SY× HZ b2 fore the component at right angles to SP is

175. A body describes an ellipse under the action of a force in a focus: find the law of force.

Let S be the focus which is the centre of force; let P and Q be any two points on the ellipse; and suppose the body to move from P to Q.

H

Resolve the velocity at P into two, one at right angles to SP, and the other perpendicular to HS; denote these by 1 and respectively. When the body arrives at Q its velocity is composed of v1 and v, parallel to their directions at P, and the velocity generated by the action of the central force during the motion, which we will denote by u. But by Art. 174 the velocity at Q can be resolved into at right angles to SQ, and , perpendicular to SH.

Hence it follows that at right angles to SP, together with u in its own direction have for their resultant v at right angles to SQ. Hence, as in Art. 33 of the Statics, the direction of u makes equal angles with the straight lines at right angles to SP and SQ, and therefore with SP and SQ. And, by Art. 38 of the Statics,

u = 2c1 sin | PSQ.

Let SP=r, and PSQ=24. Let t denote the time in which the body moves from P to Q; and let ƒ denote the accelerating effect of the force. Then if we suppose Q very near to P so that t is very small, we have u=ƒt; hence ft=20, sin p. The area described in passing from P to Q= 1⁄2ht by Art. 158; and this area may be taken to be

=

1

r2 sin 24, for it may be considered ultimately as a triangle.

where is made indefinitely small.

This shews that the force varies inversely as the square of the distance.

It is usual to denote the constant he, by μ; thus
The quantity μ is called the absolute

μ=hx

force.

ha h2a

b2

b2

176. In the preceding investigation it was shewn that the direction of the velocity u communicated by the central force while the body moves from P to Q bisects the angle PSQ. But we know by Art. 162 that this direction is that of the straight line which joins S with the intersection of the tangents at P and Q. Thus our dynamical investigation suggests that in an ellipse the two tangents from an external point subtend equal angles at a focus; and this is a known property of the ellipse.

177. A body describes an ellipse under the action of a force in a focus; required to determine the periodic time.

Let a and b denote the semiaxes of the ellipse; and h twice the area described by the radius in the unit of time. By Art. 156 the periodic time

Now it is known that the area of the ellipse is πab, and by Art. 175 we have h=

Hence the periodic time

Na

178. We can now apply the results obtained to the motions of the earth and the planets round the sun. There are certain facts connected with these motions which were discovered in the seventeenth century by the diligence of Kepler, a famous German astronomer, and which are justly called Kepler's Laws. These laws are as follows:

(1) The planets describe ellipses round the sun in a focus.

(2) The radius drawn from a planet to the sun describes in any time an area proportional to the time.

(3) The squares of the periodic times are proportional to the cubes of the major axes of the orbits.

From the second law it follows, by Art. 159, that each planet is acted on by a force tending to the sun.

From the first law it follows, by Art. 175, that the force on each planet varies inversely as the square of the distance.

From the third law an important inference can be drawn, as we will now shew. Let a be the semiaxis major of the ellipse described by one planet, μ the absolute force, T the periodic time; let a', u', T" denote similar quantities for another planet: then, by Art. 177,