148

PI. LVI.Fi S- 5 -

Astronom Y.

Point the Planet must consequently ^with the greatest Velocity, But in

tbe

poitf

re < n '

directly as those Perpendiculars, because they arvef-sely as the Velocities. .

13. If two Chords of very small Arches de> c

in the fame Time AB, B C, and DE, EF, be c ^pleated into the Parallelograms ABCV andand the Diagonals B V and E Z be drawn ; th?"those Tines tend to the 8un or Centre S, and he ^portional to the centripetal Eorce : Eor the ,j. f

and EF is compounded of B V, Be, and E Z,but B V = C f , and E Z = F/j but C c and F/ J j„generated by the Impulses of the centripetal P oiB and E, and are therefore proportional to the* 11 ’consequently so are B V and E Z. _ $

14. Draw the Diagonal AC, and it will bis e ^ ftCLine B V in h ; consequently the Sagitta B b iscentripetal Force by which the Arch ABC is def crlwhose Chord is AC.

15

about an itnmoveable Centre S, the Force in any

Hence if a Bodv revolve in any Curve

Q?> r a

any other Point p as ^-570 QB• h

for the Sagitta Q_R, q r, (whi c ^ ^

P as"

P will be to that in

to S p x X qt 1 '

s,) are as the centripetal Forces (F,_/^ if d 1?lw '‘en the Times (T, t,) are given, (by the lat t ]]S: f. But when the Forces are g[ ve! ^’

S'

is, S : r

S<‘gittæ will be as the Squares of the Times, f/S- ^ tF 5A T : 11 . Therefore when neither the Times v pjforces are the fame, it will be S : s :: F X T 2 -/

S , ...^A^

and so

T 1

S Q_P and

r-

:: F : f, And because the ellip tiC

_ _ S qp are as the Times in which they & jjvscribed, therefore when the Arches P Q_and p 4 , t.Sf

-F"

definitely small, we have T : t :: \ SP X Q/E ’

Consequently W e

qt ::SP X QT: S pXqt.

"SP' XQ.T ‘ S p*XqP

h»

l6.

I A