L x Pv as QR to Pv, that is, as PE or AC to PC;
and L x Pv to GvP as L to Gv; and GvP to
as
to
'; and (by corol. 2. lem. 7.)
the points Q and P coinciding,
is to
in the
ratio of equality; and
or
is to
as
to
, that is, as
to
or (by lem. 12)
as
to
. And compounding all thoſe ratio's together,
we ſhall have L x QR to
as
or
to
,
or as 2PC to Gv. But the points
Q and P coinciding, 2PC to Gv are equal. And
therefore to theſe, will be alſo equal. Let thoſe equals be
drawn in
and
will become equal to
. And
therefore by corol. 1. and 5. prop. 6.) the centripetal force is reciprocally as
,
that is, reciprocally in the duplicate ratio of the
diſtance SP. Q. E. I.
Seeing the force tending to the centre of the ellipſis,
by which the body P may revolve in that ellipſis.
is (by corol. 1. prop. 10.) as the diſtance CP of
the body from the centre C of the ellipſis; let CE
be drawn parallel to the tangent PR of the ellipſis;
and the force, by which the ſame body P may revolve
about any other point S of the ellipſis. if CE and
PS interſect in E, win be as,
(by cor. 3. prop,
7.) that is, if the point S is the focus of the ellipſis,
and therefore PE be given, as
reciprocally. Q. E. I.