I 5°

Astronomy.

srss ■

the Aphelion) every thing is just the revetAnd in the Points B or D it is in itsDistance from the Sun.

PI. LVI.Fig. 6.

centripetal Force tending to any given Point S. > Letbe a Tangent in the Point P, and S Y the Perp cncJ ir a

1 ^-1 tr n /•*•»!_ 1 rr , v 1 n T pM ,

P*

dio;

>s

lar, and V P the Chord passing through S. Letbe the Diameter of the Circle, and join A P. ^the Triangle S YP similar to the Triangle V AP’y ;may be sttewn from Eucl. III. 32. Therefore ^

c p v P v A (°

P V :: S P : SY; consequently-———-rS Y> aI1

SP 2 X P V 3AV~ :

AV

S Y 1 X P V, which therefore is

as

&

est

OT

is given, whence yr-p X Q_T is as Q.T ; that is,

Q.R

centripetal Force inversely ; but because AV 2 is a g' V j;>Quantity, we have the laid Force reciprocally as *

XPV3 ‘ . .

21. Again : Let it be required to find the Law °J ^

centripetal Force by which a Body is moved , so as t° g,scribe the equiangular Spiral P Q_S about the Cents? ^In this Cafe all the Angles are given in every trn*Fig. 7. Area S Q_P, and therefore also the Ratio of all the

in the Figure S P R Q_T ; therefore the Ratio

(bc-is ^ S*

Q_R iS ope

And this Ratio will be constant, let the Angle P j,changed in any Manner whatsoever : For let Q_Rwhen the Angle P S Q_is constant, and Q_T = b ’ ^jiwhen it is variable, let Q_R — x, and Q_T z^y>^_ J,(by Lem. ix. of Princip.) it will be a : x :■ b ,, b z A QT 2 Q_T* vvii*

wnence — — — — ; which shews that -gafis

a x Q- R ^ se-

al ways remain the fame as at first, viz. as S P. 1 nP . QT 2 XSP 2 .. th?

fore --Ty-g-will become S P 3 ; consequent 1 }

centripetal Force QR will be inversely as S P 3 - It

22 -

cause of the given Ratio of QT to P S) -y y— is a s

QT 2