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481.

Generally for conical revolutions, if the length of the string be l, and r the radius of the base, then the body being acted + upon by three forces ; (w) in the direction of gravity ; another (S) in the direction of the string or its tension; and the third (F) towards the centre of the base of the cone, and therefore centripetal, we have, W:S:: 1 - 7):1

1 or SE

W... ...(a)

w (e-gue) Again, by 440, the time of revolution in the circular base is

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From (c) it appears that the times of revolution, for different conical pendulums are always the same for the same altitude

( - ). In the Problem

S = 3W
in T - que = 32
and :: T =

671
Wg

which expresses the required time in seconds, since is the

2

space described in a vacuum by a falling body during that interval.

482. Generally let the perihelion distance of the comet be r, and the mean distance of the earth be a. Then, since the mean velocity of the earth is that by which a circle would be uniformly described in the time of the earth's revolution in its orbit, or in the time

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:. this velocity is

21

T Also the velocity in a parabola is generally v' =

/ 2ru
P р

р
and p = Nr?
.. at the extremity of the latus-rectum, where = 21

с

p=rv2

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483. This very ingenious theorem is easily reduced to the following ; If SAC (Fig. 97.) be a parabolic area, S being the focus, and A the vertex of the parabola, and perpendiculars bisecting AS and SC meet in H; then GH is proportional to the area ASC. For the figure being completed, we have HG+GR;RS+SM :: SC: CN SC

SC
:. HG = X (RS +

GR.
CN

.)-G

2

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there being no correction, if t is o when p=r.

But c = (274) by (453). :: the general expression for the describivg any arc of a parabola from the perihelion is

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If we use the expression (a) for the ellipse, whose equation is

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we get

będę
dt =
CN (2ap

12 – 8)
and putting

-a = 16 it becomes

b(a + u).du
dt =

cv(a*e*—u)
ae being the eccentricity.
Hence
ba
du

b

+ v (a*c* — u). ba

b

sin.-1 u

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ae

Let t = 0, when u = ae, or when e = a
ba

ba
then CE
sin.-1(-1)=

(

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Butc=b V, and u = s-a

(sin.-- + -)-> *V(c'e'-=2")

at

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and we get for the time between the vertex of the ellipse and the extremity of its latus rectum,

T= ai

(sin.-1), e) + )-1 a N (1–0)

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Again, let g = a, then the time from the perihelion to the ex. tremity of the minor-axis is

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Х

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- e). .. (e) nu Hence, and from expression () it follows that if e or the ratio of the eccentricity to the major axis is given, then the Times from the vertex either to the extremity of the latus-rectum, or to that of the axis-minor, are in the sesquiplicate ratio of the major-axes, and the subduplicate ratio of the absolute forces.

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