Book I.]

NEWTOX S PRINCIPIA.

119

OTHERWISE.

By Lemma X, Cor. 4,

„ space ipso motus initioA< a t 2

„ sa S-

To generalize this expression, let be the space described in 1" at

the surface of the Earth by Gravity. Also let the unit of force be Gravi-ty . Then

F : 1

t* ‘ 2xT

.-. F

by hypothesis.

2 saar.

g t s

— X

(a)

187. Cor. 1. F a

Q R

oc

Q R

(area S P Q)

a

Q R

S P 2 x Q T 2 '

To generalize this, let a be the area described in 1 ". Then the areadescribed in t" = a X t = S - P X Q T .

.-. t =

_ SP x QT

2 a

and substituting in (a) we get

F = 8 a ■ x

Q R

g SP 2 x QT 2 • • • •Again, if the Trajectories turn into themselves, there must bea : 1" : : A (whole Area) : T (Period. Time)

A

.-.a — >jr.

Hence by (b) we have >

QR

(b)

W _ 8 A ' v,

* ~ g T* X SP ! xQT !

which, in practice, is the most convenient expression.

1Qa n a t? 8 A 2 Q R

. Cor. 2. F - gX2 X SY 2xQP 2 ‘ ’

1

( C J

(d)'

139. Cor. 3. F = x

g 1 2

s y 2 x py

• • (e)