Sidebilder
PDF
ePub

Again, in expression (d)

let e = 0. Then

Te

.

2

Ne
and 4T =

2mat
NH

(9)

the Periodic Time in a circle when the force ce this also follows

from (f).

Again in the expression (d), if 4r be the latus-rectum, we have

[merged small][ocr errors]

12

X

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

[ocr errors]

2

Te

(1 - 1)

[merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small]
[merged small][merged small][ocr errors][ocr errors][merged small][ocr errors]

which is the same as (c).

This last case affords a striking example of the utility of the Theory of Vanishing Fractions.

1
485. The angular velocity of (463)

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

1

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
::F=

9
R

[merged small][ocr errors][merged small][merged small][merged small][ocr errors][merged small][ocr errors][merged small][merged small][merged small][merged small]

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.

[merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][ocr errors][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][ocr errors][subsumed][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

Also since

pip:: gd : de from similar triangles in the elemental figure, we have

Rde § V (nR

n-1.6)

do =

1

[ocr errors]
[ocr errors]

1

{+

[ocr errors]

...V (nRo-n–1.5") - Run + c 2 n

(nR'-n-1.8") +RWN
see Hirsch's Integral Tables, p. 122.
Let 6 = 0, when ç= R, and we have
Ꭱ - Ꭱ !n

1 ✓n
C=
R+ Rn

It Nn ..0 =

1+wn .1.

(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

1

[ocr errors]
[ocr errors]

2

[ocr errors]

way to the centre.

In this case by (a)
R

R
p=
2w (n 1

(3n + 1)

4
Hence, the velocity at that point is

V=L=H x V3n + 1
р

R

R and the velocity in a circle at the distance is

[merged small][merged small][merged small][ocr errors][merged small][merged small]

If the force be the same in the circle and parabola,

488. we have

[merged small][merged small][merged small][ocr errors][merged small][ocr errors][merged small][merged small][merged small][ocr errors][merged small]

489.

Let t be the time required, and make F=. Then

[ocr errors][merged small][merged small][merged small]

a being the distance from the centre when the body begins its descent.

Letę = a
a being the given space. Then

vo = 2 x l.

a

[ocr errors]

which gives the velocity acquired down the given space.

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][ocr errors]
« ForrigeFortsett »