inually drawing it towards _S_, and so it really finds
itself at _P_, having described the circular arc _OP_, which may
be considered to be compounded of, and analyzable into the
rectilinear motion _OA_ and the drop _AP_. At _P_ it is for an
instant moving towards _B_, and the same process therefore carries
it to _Q_; in the third second it gets to _R_; and so on: always
falling, so to speak, from its natural rectilinear path, towards
the centre, but never getting any nearer to the centre.

The force with which it has thus to be constantly pulled in towards
the centre, or, which is the same thing, the force with which it is
tugging at whatever constraint it is that holds it in, is
_mv^2/r_; where _m_ is the mass of the particle, _v_ its
velocity, and _r_ the radius of its circle of movement. This is the
formula first given by Huyghens for centrifugal force.

We shall find it convenient to express it in terms of the time of
one revolution, say _T_. It is easily done, since plainly T =
circumference/speed = _2[pi]r/v_; so the above expression for
centrifugal force becomes _4[pi]^2mr/T^2_.

As to the fall of the body towards the centre every microscopic
unit of time, it is easily reckoned. For by Euclid III. 36, and
Fig. 58, _AP.AA' = AO^2_. Take _A_ very near _O_, then _OA = vt_,
and _AA' = 2r_; so _AP = v^2t^2/2r = 2[pi]^2r
t^2/T^2_; or the fall per second is _2[pi]^2r/T^2_,
_r_ being its distance from the centre, and _T_ its time of going
once round.

In the case of the moon for instance, _r_ is 60 earth radii; more
exactly 60.2; and _T_ is a lunar month, or more precisely 27 days,
7 hours, 43 minutes, and 11-1/2 seconds. Hence the moon's
deflection from the tangential or rectilinear path every minute
comes out as very closely 16 feet (the true size of the earth being
used).

Returning now to the case of a small body revolving round a big one, and
assuming a force directly proportional to the mass of both bodies, and
inversely proportional to the square of the distance between them:
_i.e._ assuming the known force of gravity, it is

_V Mm/r^2_

where _V_ is a constant, called the gravitation constant, to be
determined by experiment.

If this is the centripetal force pulling a planet or satellite in, it
must be equal to the centrifugal force of this latter, viz.