the Periodic Time in a circle when the force c

from (f).

(9)

— this also follows

Again in the expression (d), if 4r be the latus-rectum, we

Now when the Ellipse becomes a Parabola, e = 1. Consequently the time from the vertex to the extremities of the latus rectum in the Parabola is

when c = 1, and we have

This last case affords a striking example of the utility of the Theory of Vanishing Fractions.

485.

The angular velocity ∞ —— (463)

.. if v and v' denote the velocities of the earth at its mean distance a and perihelion distance a - ae, ac being the eccentricity, we have

486.

The centripetal force acting upon a point placed within a sphere oc g, see Newton's Prop. LXXIII. Consequently, if g be the force at the surface of the sphere, and R its radius, we have

But the velocity (V) in a circle at the Earth's surface is that which

R

would be acquired down

with the force considered constant.

from similar triangles in the elemental figure, we have

see Hirsch's Integral Tables, p. 122.

Let = 0, when હૃ R, and we have

(1+√n ̧ √(nR3—n— 1. ç3) - R√ n ) .......(b)

√ (nR3 — n − 1 . ç3)+R√/n)

the polar equation of the spiral.

Let = 0. Then
ę0.

or the number of revolutions will be infinite before the body falls into the centre, that is, it never reaches it, although it continually approaches the centre.

488. we have

If the force be the same in the circle and parabola,

489.

Let t be the time required, and make F, Then

a being the distance from the centre when the body begins its descent.

which gives the velocity acquired down the given space.