p =

R3 x density gR2%, which is therefore known.

203
√(M+m)

and M + m =

4a3

3

p2

Generally when the force acts in parallel lines

473.

day

F=

dx2

y being parallel and x to the direction of the force.

But in

But in the circle it will be likewise found that

Hence if the absolute force be changed from μ ao to μ' =

an ellipse having the same diameter 2a as the circle, and any

CENTRAL FORCES.

axis-minor whatever may be changed; and generally if it be required to ascertain the change in the absolute force requisite to make the body describe an ellipse of given axes 2a', 26′, we have b'4

a

and at the vertex and extremity of latus rectum is respectively

475.

Since the body revolves in an hyperbola with the force in the centre, we know from Newton, Prop. X., or very readily from

Now the mean angular velocity is that by which a body would uniformly describe a circle in the periodic time of the earth,

and at the mean distance a, the angular velocity is

478.

Q. E. D.

Generally, let R be the given distance or radius

vector, P the corresponding perpendicular upon the tangent, then

which is the equation to a conic section, and therefore the orbit has one apse at least. But the problem must be solved without considering the nature of the curve; for since at an apse

and will therefore have at least two real values, which indicate

two apses.

If the velocity of projection be given and equal to ; then

1

Vel. in parab.: vel. in circle same dist. :: √2
and vel. in s. dist. : vel. in ellipse :: √a: √2u

.. vel. in parab.: vel, in ellipse :: √2a: √2a - e

and it may easily be shewn that in the parabola, whose latus rectum is 4r,