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Again, in expression (d)
let e = 0. Then
the Periodic Time in a circle when the force ce this also follows
Again in the expression (d), if 4r be the latus-rectum, we have
and T = 212. r
2 -e (1 – e) – sin.-1e Nu
(1 – e?) 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
(1 - 1)
which is the same as (c).
This last case affords a striking example of the utility of the Theory of Vanishing Fractions.
8 e :. if v and v' denote the velocities of the earth at its mean distance a and perihelion distance a ae, ae being the eccentricity, we have v: 0 ::
:: Ime: 1.
Q. E. D. aa a? (1
486. The centripetal force acting upon a point placed within a sphere of g, see Newton's Prop. LXXIII. Consequently, if g be the force at the surface of the sphere, and R its radius, we have
F: 9:: 5 : R
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.
pip:: gd : de from similar triangles in the elemental figure, we have
Rde § V (nR
...V (nRo-n–1.5") - Run + c 2 n
It Nn ..0 =
✓ (nRe-n-1.5") – RJ n ......(6) 2n
w (nR-n-1.8")+Rvn) the polar equation of the spiral.
Let <= 0. Then
log. 0 = 0.
2n 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.
R Again, let g = then the body will have approached half
way to the centre.
In this case by (a)
♡ (3n + 1)
V=L=H x V3n + 1
R and the velocity in a circle at the distance is
If the force be the same in the circle and parabola,
488. we have