Mech a n I c s.

But is the Velocity be given, it will be as the Dijlnr.ee

V z , i

from the Centre inversely , (for then — becomes—.)

J a a

5. Hence if the Central Forces in two Circles beequal, the Diameters of those Circles , or Dijlavces fromthe Centre of Force, will be as the Squares of the Velo-cities direflly.

6. As AE : 1 : : 3,14160 : P — the Periodical Timepf describing the whole Circle (for in equable Motions

the Spaces are as the Times.) Therefore P — ? ■ ’- ° ;

whence V 1 =

3,1416* X o z

F :

which substituted in the

Expression for the Central Force, we have

3,i4i6 z x aP z

a 7—2

— p- t (because 3,1416 is given) for the Central Force

in this Case ; which therefore is always as the Diameterer Radius of the Circle directly, and the Square of thePeriodical Time inversely.

7. Hence if the Central Force be as any Power (n)of the Distance ( a ) from the Center C, we have ^

a"; whence P — --- ; and therefore when n—o,

a 2

we have P — ah, or the Periodical Time will be as theSquare Root of the Dijlance.

8. If n — 1, that is, if the Force be directly as the

Distance ; then will P — a — — a 0 — 1 ; that is, the* 2

Periodical Time will be given, or the fame in every Circle.

9. If n — 2, that is, if the Force be directly as

the Square of the Distance, then P — a s — —— ;

o*

or the Periodical Time will be as the Square Root of theDijlance inversely.

10. If n — 1, that is, if the Force he inversely

as