from whence, by means of the equation to the given curve, z may

be found.

Ex. 1. In the Problem it is required to find 2 for the common Parabola.

2

+

a

a

(3). Required the space fallen through to acquire m timas the velocity in a curve, when

is easily shewn from expressions (d) and (ƒ) that

438. It is well known (see Vince's Fluxions, Simpson's, &c., that the velocity of a body at any point of its orbit, is the same as

would be acquired through

point, with the force constant.

But this chord is (see Vince).

2pdę dp

the chord of curvature to that

which gives the general relation required.

To apply this to the conic sections, we have

for the Parabola, Ellipse, and Hyperbola respectively. Hence

439. Suppose a body moving in a curve PQ Fig. 91.; its motion at P is in the direction of the tangent PR, and if PR described in a given time be taken to represent the velocity v, and be resolved into Pr, Rr, then Pr is the velocity with which it is approaching the centre S. Again QR=Tp is due to the force, and we therefore have, when there is no angular velocity, the whole approach to the centre = PT. But with this angular velocity the approach is only

SQ

Sp. Hence Tp must be the recess from the centre caused

being the centrifugal force; which is the relation required. To apply it to the hyperbolic spiral, we have

440.

and F9.

=

p3

The velocity in a circle is that which is due to

chord of curvature or to its radius.

.(b)

Now at the surface of the earth F

=

-32

feet nearly.

6

Hence at the distance of n times the earth's radius R from its

feet, or g into miles, and the problem will be resolved.

Suppose it were required to find the velocity and periodic time

of the moon; then we have n = 60 and

= .6356 miles in a second nearly.

and P237235 seconds nearly.

=27 days, 10 hours, 59 seconds, nearly.

which is sufficiently near for a rough calculation, the sidereal revolution requiring 27 7 43′ 11′′.5.

441.

Generally let the comet be supposed to change the

velocity of the earth from v to mxv.

Now let first be investigated the nature of the orbit, from having