Hence, since by the question TT, we have

By Prop. VII. cor. 2, of Newton's Prin. we have (see Newton's

Fig.)

Fat R: Fat S :: RP2 x SP: SG3.

Now, let S be in the centre of the circle, and R be in the circumference; then we have

But if and be the absolute forces, we have

529.

The velocity in an ellipse at any point of it where

the on the tangent is p, is

P

c being = b√√✓ (453). Hence at the mean distance, where pb, we have

where c' (considering the O a limit of an ellipse) is. Hence

Hence if the absolute forces, p, p, be the same in the circle and ellipse, and the radius (r) of the circle be equal to the semimajor-axis (a) of the ellipse, we have

or the velocity in the ellipse at its mean distance is the same as that in a circle whose radius is the mean distance, and centre of force the same as that in the ellipse.

Q. E. D.

Again, in the ellipse the periodic time is (484)

v2 = 2μ ƒ −ędę = μ. (a2 — p2)

a being the whole distance to the centre.

Hence the velocity acquired in falling to the centre is

Again, if the force at the beginning of the descent, viz. μa, be considered constant, the velocity acquired down the whole distance with half this force is

μα

√2 × Fx dist. = √ 2. xa = a√μ

2

V' =

531.

CENTRAL FORCES.

If R and r be the radii of the globe and wheel of the epicycloid, the equation referred to the centre of the globe is easily proved to be

532. Let CA or CB = a, and make CD = b. since DE = a, by the hypothesis we have

CE = √ a2 — bo.

Now the force at C varying as the distance, let it be

Then

Hence the velocities of the balls when they arrive at D and E are respectively

v = √ μ. √ (a2 — 6o), and v' = √μ √ (ao! — a2 — b2) = √ μxb,

Now the bodies being perfectly elastic, move after impact with the planes with the same velocities as they impinge.

are

Therefore the initial velocities at the distances b and √a2 – b2

√μ. √(a3 — b3), √ μ.b

and since the directions of the initial movements, or tangents of the orbits, are inclined to the distances at angles of 180° — 45° ; .. the initial perpendiculars on the tangents are respectively

PR. sin. 45° and P'R' sin. 45°

Again, since the force varies as the distance the orbits are ellipses with the force in the centre, and the equation to the ellipse referred to its centre is

:: aß b √√a2 = b2, and a'f' = b √a2 = b2.

2

Hence, and from Equat. (1) we have

and from these four equations it is easy to find by the solution of quadratic equations the values of a, B; a', B'; and therefore to construct the orbits.

533. Let a be the distance of the earth from the Moon, their quantities of matter; then if be the distance from the earth at which their attractions are equal, we have

'?

Again, if r, r' be any distances from the earth and moon of the body in the curve of equal attraction, we have

But if x and y denote the rectangular co-ordinates of the required curve originating in the earth's centre, and measured along the line joining the earth and moon, we have

consequently the curve is a circle whose radius is αν (μμ)

534.

If F, F' denote the forces, v and v' the velocities,

and R, R' the distances; then since

v v Fx R: Fx R'

But the velocity in any curve at any point of it is equal to that

which would be acquired down chord of curvature at that point.

4

Hence if c, c' denote these cho.ds at the points of projection, we have