Review of the Principia of Newton. 331 
te lf a body be acted on by a eons force only, which 
ts constant, it will descend with an accelerated motion in a 
right line toward the centre of force, 5 is the case of edics 
—s near the surface of the eart 
. The force is proportional to the in increments of t r : 
ies or the accelerations of the motion are piaporiinal % 
their cause, viz. the intensity of the force 
3. If the body have a motion of projection in any oe 
direction, with the line of its rectilinear descent, and: if t 
centripetal force vary in the inverse duplicate ratio of the 
distance, it will move in some one of the conic sections, hav- 
ing the centre of force in the focus of the figure. 
he species of the conic section, and the particular 
curve of that species, which the body will describe, de 
on ihe ratio of centripetal force to the velecity of projection. 
If the force be uniform and constant, the veloci- 
ty pase for a movement in any of the conic sec- 
tions, will be that which is due to 4 of the latus rec- 
direc 
tion of the Sewn ones force. When, therefore, that height 
s 4 of the point of projection the 
eeatias of force, it t will move in a circle, whose center will be 
the seat of force: If that height be less, it will move in an el- 
— within the circle, and the focus of the ellipsis farthest 
m the point of projection will be the seat of the force. It 
will continue to be an ellipsis until the velocity of projection, 
and consequently 4 of the Jatus rectum=0; it will then be- 
= a straight line. 
. If the height due to the pemoeey be greater than that 
anameial a circle, it will still move in an ving 
the centre Saccee i its nearest focus, until that height reach 
the limit of the ratio sa Fg 1, of that of the circle, in w 
case the body will move in a parabola; and if the velocity of 
a ea be still greater, it will move in an hyperbola. 
he periodical times of bodies moving in all ellipses, 
Sant major axes are equal, are the same, sodas al to that 
is 
g. The periodical times of bodies moving about the same 
center, “a different elliptical or circular orbits, will be in 
the dae — of their — axes, or diameters. 
The velocities of ving about the same center, 
in Sie orbits, are in the inverse sla. aelthsplicthe ratio of their 
distances in similar positions of their orbits. 
