which is the velocity required, when the direction of the impulse is any whatever.

When the impulse is made in the direction of the body's motion, we have

P = R

and m =

± √ {1 ± √(1 R. 1 - c2)}.

461.

This is true for all the planets. For generally

and p decreases continually from the mean to the perihelion distance.

462.

Let a be the given distance from the plane; then

Also let V be the velocity of projection, a the inclination of its direction to the horizon; then V. cos. a is the velocity parallel to the horizon

=

or

But

√(ay — y3)

fন

ydy

=

√(ay — y3)

1

2μ
V COS.a

.dx

2μ
dx

S

√ay-y2

Vcos. a

== √(ay — y) + a f

- √ (ay—y3) + 4/ X vers.-1 2y + c

dy

and we get

which is the equation to a quasi-cycloid, the diameter of whose ge

nerating circle is a.

vers.-1

(ay y

a

463.

we have

But c = b

If denote the angular velocity of the radius rector,

a

a. (1-e)

1 + e cos. 0

464. The equation to the circle referred to a point in the circumference is

=2r sin. 0

when is measured from the tangent at that point.

Again, when force is in the centre, we have (440)

465.

Since the body is as much retarded in its whole ascent as it is accelerated in its descent, the time of revolution is the same as that of a body moving in a circle with the uniform velocity

Again, the time of an oscillation in a very small circular arc, whose radius is 1, is

T = T

..TT: m: 1.

See 570.

466.

Since the force is repulsive, the trajectory will be convex to the plane, and its plane passing through the line of projection will be to the given plane, there being no reason why it should be inclined on one side rather than another.

Hence y, x denoting the co-ordinates and parallel to the plane, a being 0 when y = a the given distance from the plane, we have

F=

d'y =

and consequently the required Trajectory is an Hyperbola whose semiaxes are

a2ß and a

respectively.

467.

Let PQ (Fig. 96,) be the orbit of the Earth round the Sun, S, and pq that of the Moon round the Earth P; then if P, p be their positions when in conjunction, and Q, q those of the next instant; their orbit in fixed space, viz., the Epicycloid, will then be convex or concave to the Sun, according as

QR-qr

is positive or negative.

But if P, p; R, rare the respective known Periods and Distances of the Earth and Moon, we have (Princip. prop. IV.)