for the Integration of which see Lacroix's Differential and Integral Calculus, 4to edit., or Whewell's Dynamics, p. 15.

490.

The tension is the centripetal force necessary to

retain the stone in the circle, or it is

and .. by substituting and performing the arithmetical operations we shall get the time required.

491.

Generally let it be required to find the velocity acquired through (n + 1) x R to the earth's centre; R being its

the velocity acquiring in descending to the surface of the earth.

the velocity acquired from the surface to the centre.

492.

Let a and b be the major and minor axes of the

given ellipse. Then the greatest dist. = a + √ a2-b2= a(1+e)

is known.

Again, the Periodic Time in the Ellipse is (484)

and in the circle for the same force it is (considering it the limit

493.

and

By (443) if v and v' denote the angular velocities of

VOL. II.

=

告

1 αμ Χ

p3 × (2a - p) 3

при

or the angular velocity of p in the ellipse, the force being in the focus, is least at the extremity of the minor-axis.

494. By 487 when from the force suddenly becomes

we have

Rp

√ (aR2-n-1.p2)

P=

24n

1.

· . ¿. {1 +√n ̧ √(nR2 — n— 1 po)—R√n? [ 1 — √ n ` √ (nR2 — n−1.g')+R,√nS

Hence, if a few terms of this converging series be summed, and the sum be called S, we get

which may be arithmetically computed to any degree of accuracy, by means of Logarithmic Tables. The whole number in the result will answer the problem.

Let a, b be the semiaxes of the Ellipse; then since

495.

(by 436)

a + √a2 - b2, which is the distance from the farther apside,

see Hirsch's Integral Tables, pp. 141 and 140.

Let ęr; then

Lete

apse is

T=

2

0. Then the time to the focus from the farther