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Also by 484, the Periodic Time in the Ellipse is

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496. The time from Perihelion to mean distance is (by 484 . e)

3
a?
Х

e)
Tu
where

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:. the time from mean distance to mean distance through aphelion is

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Now if be the radius required, since the Periodic Time in the circle is

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we have
Udo =

mpde and v2 = v(a? - $). let ę = 0; and the velocity acquired through a, or the velocity of the system is

V = av de Again, if a - p = x be the space described I direction of the motion of the system, whilst the body acquires the velocity v, we have dx

dx dt =

7 px (2ax-x2) But since the motion of the system is uniform, if dy denote the distance moved through in the time dt, we have

dy = V x dt ad re x dt
dy

dx
dt =
adu

Nr. 7 (2ax– x2)
dix

xo)
But when x = 0, y = 0, and . C = 0.

Hence

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3 vers.

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+ C.

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the equation to the curve traced in the plane passing through the two directions of motion. Hence the curve is a sinuoid, which may easily be constructed by the formula tan. () =

dy

dr (2ar - .) and the equation (m); Q being the inclination of the tangent at any point to the line of abscissæ.

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Let a

498.

p be the altitude fallen through in the time t, a being the finite distance from the centre at which the body begins its descent; then, by the question

(a – p)" = M.(a – p)" i. dt = nM. (a –p)–1.dz =

de

V

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:

499. If F and denote the centripetal and centrifugal forces respectively, we have, by 439, F: 0 ::

de p3

dp
d


dp
But F

cdp

(436) podę

х

X F.

3

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But if V and P be the initial velocity, and perpendicular upon the tangent, at any given point, we have

V
P

Vo
and .. p =

(a) P.pt

Hence for different parts of the same trajectory, we have

(6)

1

1

for different trajectories, having at the same one point of them the same velocity V

.. (c) Papel for different trajectories having for same one point of them the same I upon the tangent P

V
OC

..(d)

and for different trajectories, having neither a common velocity por a common I

upon

the tangent V?

.. (e)

P.

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62

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Again, in the circle, p=g, and if p = a - ae = the perihelion distance, the velocity in the circle is

Ć

c'

a.(1-e) V

w(1+e?)

웅 Х v

ite

V =

But if he be the absolute force we have generally in the ellipse whose axes are 2A, and 2B (see 453,)

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and for the circle (considering it the limit of an ellipse)

Ćeva.(1 - e)s.

= V (1+e) x V (1+e?)

Hence
V

(

ite v (iter)

(6) (1+e) New let e become Indefinite, and let L be the Limit; then putting

у

= L +1

V l being the Indefinite part of the ratio, we have

it es (L +178 =

1 te or L+ 2L1 + 18 + e(L + 1) = 1 + es and equating Definite Quantities, (see Wright's Commentary on the Principia of Newton, sect. 1.)

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