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Fig. 2.

D

Let F

of the circle.

V
R

S

centripetal force at A, (fig. 1,) tending to the centre C

velocity of the body.

radius AC of the circle.

P = periodic time.

Then (Art. 4,) Ptv

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2R

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But s is as the force F that generates

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; and, since 2, #, and t, are given quan

9.-Cor. 1. The periodic times are as the radii directly, and the velocities reciprocally.

For (Art 4,) V✔2Rs = √/2RF, and V2 = 2RF, and P = ≈t √

2R

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2R and PP72 12 × therefore P2 V2 = m2 t2 × 4R2, and P2 = F' πt × 2R R

F'

10.-Cor. 2. The periodic times are as the velocities directly, and the centripetal forces reciprocally.

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10. Cor. 3. If the periodic times are equal, the velocities, and also the centripetal forces, will be as the radii.

R

R
and and

V

For, if P be given, then F V' F

are all given ratios.

12.-Cor. 4. If the periodic times are as the square roots of the radii, the velocities will be as the square roots of the radii, and the centripetal forces equal.

For (Art.8,) putting ✔R for P, we have ✔R α✅

VF

R R

α Therefore, 1 X

F

α, and Rα V, and ✔F is a given quantity.

13.-Cor. 5. If the periodic times are as the radii, the velocities will be equal, and the centripetal forces reciprocally as the radii.

R R

For, putting R for P, we have R x; whence Rα

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that is, Rα, or the centripetal force is reciprocally as the radius;

and V is a given quantity.

14.-Cor. 6.-If the periodic times are in the sesquiplicate ratio of the radii, the velocities will be reciprocally as the square roots of the radii, and the centripetal forces reciprocally as the squares of the radii.

Put R for P, then Rαα I; and Rα

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R R
F

1

F

; or RR α, and ✔R

15.-Cor. 7. If the periodic times be as the nth power of the radius, then the velocities will be reciprocally as the n-1th power of the radii, and centripetal forces reciprocally as the 2n-1th power of the radii.

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16.-If several bodies revolve in circles round the same or different centres, the velocities are as the radii directly, and periodic times reciprocally.

For, putting the same letters as in Art. 8, we have (Art. 4,) V =

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V

V

V2RS V2RF; and P x

F

(Art. 10,) and PF ∞ V, and Fαp

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17.-Cor. 1..The velocities are as the periodical times, and the centripetal

forces.

For we had PF α V.

18.-Cor. 2. The squares of the velocities are as the radii and the centripetal forces.

For V2RF.

19. Cor. 3. If the velocities are equal, the periodic times are as the radii, and the radii reciprocally as the centripetal forces.

R
P

For, if V be given, its equal is a given ratio; and ✔RF is given, whence

Rα.

20. Cor. 4. If the velocities be as the radii, the periodical times will be the same, and the centripetal forces as the radii.

R
P'

For then V or R α and 1 α: ¤ Also R = √2KF, whence Rα F.

21. Cor. 5. If the velocities be reciprocally as the radii, the centripetal forces are reciprocally as the cubes of the radii, and the periodic times as the squares of the radii.

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for V, then (Art. 18,) 1 =✔RF, =2RF, whence Fx

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Also, and P & RR.

22.-If several bodies revolve in circles about the same or differ. ent centres, the centripetal forces are as the radii directly, and the squares of the periodic times reciprocally.

Put the same letters as in Art. 8. Then (Art. 4,) P = =t

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23. Cor. 1. The centripetal forces are as the velocities directly, and the periodic times reciprocally.

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R V
PP p.

24.-Cor. 2. The centripetal forces are as the squares of the velocities directly, and the radii reciprocally.

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R

and FP V. But (Art. 9,) Pα, therefore FP α

therefore XV, and FX

VV

R

25.-Cor. 3. If the centripetal forces are equal, the velocities are as the periodic times; and the radii as the squares of the periodic times, or as the sqnares of the velocities.

26.-Cor. 4.—If the centripetal forces be as the radii, the periodic times will be equal.

1

For Fα

R
PP'

F 1 and α PP

F and if R

be a given ratio, will be given, as

PP

also P.

27.-Cor. 5. If the centripetal forces be reciprocally as the squares of the distances, the squares of the periodical times will be as the cubes of the distances, and the velocities reciprocally as the square roots of the distances.

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28. If several bodies revolve in circles about the same or different centres, the radii are directly as the centripetal forces, and the squares of the periodic times.

For, (Art. 4,) putting the same letters as before, Ptv

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29.-Cor. 1. The radii are directly as the velocities and periodic times. For (Art. 17,) PF ∞ V, but PPF R; therefore PV & R. 30.-Cor. 2. The radii are as the squares of the velocities directly, and the centripetal forces reciprocally.

V

For (Art. 10,) Pα; but (Art. 29,) Ra PV; therefore Rα

F

VV
F

31.-Cor. 3. If the radii are equal, the centripetal forces are as the squares of the velocities, and reciprocally as the squares of the periodic times; and the velocities reciprocally as the periodic times.

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SCHOLIUM. The converse of all these propositions and corollaries are equally true; and, what is demonstrated of centripetal forces, is equally true of centrifugal forces, they being equal and contrary.

32.-The quantities of matter in all attracting bodies, having others revolving about them in circles, are as the cubes of the distances directly, and the squares of the periodical times reciprocally.

Let M be the quantity of matter in any central attracting body. Then, since it appears, from all astronomical observations, that the squares of the periodical times are as the cubes of the distances, of the planets, and satellites from their respective centres. Therefore (Art. 14,) the centripetal forces will be reciprocally as the squares of the distances; that is, Fα And (Art. 1,) the attractive force,

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RR

at a given distance, is as the body M; therefore, the absolute force

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33.-Cor. 1. Hence, instead of F in any of the foregoing propositions and

M

their corollaries, one may substitute RR'

ing body in C exerts at A. (Fig. 1.)

which is the force that the attract

34.-Cor. 2. The attractive force of any body is as the quantity of matter directly, and the square of the distance reciprocally.

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The Motion of Bodies in all sorts of Curve Lines.

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35. The areas which a revolving body describes by radii drawn to a fixed centre of force, are proportional to the times of description, and are all in the same immoveable plane.

=

Let S (fig. 3,) be the centre of force; and let the time be di Ivided into very small equal parts. In the first part of time, let the body describe the line AB; then, if nothing hindered, it would describe BK AB, in the second part of time; and then the area ASB BSK. But, in the point B, let the centripetal force act by a single, but strong, impulse, and cause the body to describe the line BC. Draw KC parallel to SB, and complete the parallelogram BKCr, then the triangle SBC = SBK, being between the same parallels; therefore SBC SBA, and in the same plane. Also, the body moving uniformly would, in another part of time, describe Cm CB; but at C, at the end of the second part of time, let it be acted on by another impulse, and carried along the line CD; draw mD parallel to CS, and D will be the place of the body after the third part of time; and the triangle SCD SCm = SCB, and all in the same plane. After the same manner, let the force act successively at D, E, F, &c. And, making Dn = DC, and Eo = ED, &c. and completing the parallelograms as before, the triangle CSm CSD DSn = DSE = ESO ESF, &c. aud all in the same immoveable plane. Therefore, in equal times, equal areas are described; and, by compounding, the sum of all the areas is as the time of description. Now, let the number of triangles be increased, and their breadth diminished ad infinitum, and the centri. petal force will act continually, and the figure ABCDEF, &c. will become a curve; and the areas will be proportional to the times of description.

36.-Cor. 1. If a body describes areas, proportional to the times, about any point, it is urged towards that point by the centripetal force.

For a body cannot describe areas, proportional to the times, about two different points or centres, in the same plane.

37.-Cor. 2. The velocity of a body revolving in a carve is reciprocally as the perpendicular to the tangent, in that point of the curve.

For, the area of any of these little triangles being given, the base (which represents the velocity,) is reciprocally as the perpendicular.

38.-Cor. 3. The angular velocity at the centre of force is reciprocally as the square of its distance from that centre.

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