force which acts in the line joining the particles, and varies inversely as the square of the distance.

The law thus suggested is assumed to be universally true, and calculations are made of the effects of the action of the bodies of the solar system upon one another in disturbing their elliptic motion; and also of the disturbances of the motion of the satellites due to a want of exact sphericity in the primaries; and these calculations have been found to agree with the results of most minute astronomical observations.

Predictions of the return of comets have been fulfilled, founded on the supposition of the truth of the law, and the existence and position of a planet have been recognized, before its discovery by actual observation, from its assumed action according to this law upon another planet.

Thus the law of gravitation has satisfied every test which has hitherto been applied to it, and it is so far proved to be true where our system is concerned.

PROP. XVI. THEOREM VIII.

On the same supposition, the velocities of the bodies are in the ratio compounded of the inverse ratio of the perpendiculars from the focus on the tangent and the subduplicate ratio of the latera recta.

COR. 1. The latera recta of the orbits are in the ratio compounded of the duplicate ratio of the perpendiculars and the duplicate ratio of the velocities.

72

For L: L':: h: h"" :: V. SY: V". SY

COR. 2. The velocities of the bodies, at their greatest and least distances from their common focus, are in the ratio compounded of the ratio of the distances

inversely, and the subduplicate ratio of the latera recta directly.

For the perpendiculars on the tangents are these very distances.

COR. 3. And therefore the velocity in a conic section, at the greatest or least distance from the focus, is to the velocity in a circle at the same distance from the centre in the subduplicate ratio of the latus rectum to twice that distance.

For the latus rectum of a circle is the diameter, therefore if SA be the greatest or least distance, velocity in the conic section: velocity in the circle

COR. 4. The velocities of bodies revolving in ellipses are, at their mean distances from the common focus, the same as the velocities of bodies revolving in circles at the same distances; that is (by Cor. 6, Prop. iv.), in the inverse subduplicate ratio of the distances.

For the perpendiculars are now the semiaxes minor, that is SY-BC, and the distance SB = AC, therefore velocity in the ellipse at the mean distance: velocity in the circle at the same distance

COR. 5. In the same figure, or in different figures having their latera recta equal, the velocity varies inversely as the perpendicular from the focus on the tangent.

COR. 6. In the parabola, the velocity varies in the inverse subduplicate ratio of the distance of the body from the focus, in the ellipse it varies in a greater, and in the hyperbola in a less inverse ratio.

COR. 7. In the parabola, the velocity of the body at any distance from the focus is to the velocity of a body revolving in a circle at the same distance from the centre, in the subduplicate ratio of 2:1; in the ellipse it is less, in the hyperbola greater than in this ratio.

For, velocity in the conic section velocity in the circle at the same distance

I. (2SP) * .. (L.SP)

SY SP

::

:: √21 in the parabola,

: 1 :: (HP)*: 1 in the

AC

: 1

1 in the ellipse or hyper

bola, and HP<2AC in the ellipse, and >2AC in the hyperbola.

Hence also, in the parabola, the velocity is everywhere equal to the velocity in a circle at half the distance, in the ellipse less, and in the hyperbola greater.

COR. 8. The velocity of a body revolving in any conic section is to the velocity in a circle at the distance of half the latus rectum, as that distance is to the peprendicular from the focus on the tangent. For, the velocity in the conic section: the velocity in L T the circle at distance L :: SY L :: L: SY.

COR. 9. Hence, since (Cor. 6, Prop. IV.) the velocity of a body revolving in a circle is to the velocity in

any other circle in the inverse subduplicate ratio of the distances, the velocity of a body in a conic section will be to the velocity in a circle at the same distance as a mean proportional between that

common distance and half the latus rectum to the perpendicular from the focus on the tangent. For velocity in a circle at distance L: velocity in a circle at distance SP :: SP: (L), therefore velocity in conic section velocity in circle at distance SP :: (¿L. SP)* : SY.

Notes.

223. To find the velocity in a conic section described under the action of a force tending to the focus.

or else, V2= F.1⁄2PV=

μ. HP

==

SP AC SP.ACʼ

but HP-2AC - SP in the ellipse,

and, HP SP-2AC, in the hyperbola, force repulsive,

=

=

SP+2A C, in the hyperbola, force attractive;

velocity in the ellipse reduces itself to that for the hyperbola under an attractive force by changing the sign of CA, which corresponds to the opposite direction in which AC is measured in the hyperbola; it is reduced to that for the hyperbola under

H H

a repulsive force by changing the sign of μ, which corresponds to changing the direction of the force; and to that for the parabola by making AC infinite.

225. To compare the velocity in the ellipse or hyperbola with that in the circle at the same distance.

Let U be the velocity in the circle,

226. DEF. If from any point lines be drawn representing in direction and magnitude the velocity of a particle describing an orbit under the action of a force tending to a fixed centre, the locus of the extremities of these lines is the Hodograph.

This name is given to the curve by Sir William Hamilton, in his work on Quaternions.

227. Since the velocity in a central orbit is if SQ be

h taken in SY equal to SY'

SY' the locus of Q will be the polar reci

procal of the orbit with respect to a circle, the square of whose radius is h; and if it be turned about S through a right angle will be the hodograph of the orbit.

228. If a conic section be described under the action of a force tending to a focus, the hodograph will be a circle.

For, in the case of an ellipse or hyperbola, the velocity varies inversely as SY, and therefore directly as HZ, to which its direction is perpendicular, and the locus of Z is a circle. And, in the case of a parabola, AY being the tangent at the vertex, AU perpendicular to SY,

SY: AS:: AS: SU,

therefore SU varies as the velocity, and the locus of U is a circle which passes through S.