Centripetal Forces. g - 
and pm be perpendicular to SL ; then, if the velocity of the 
body falling in the right line SL at the point M be kY, the 
velocity of the body at the point m acted on by the above men- 
tioned forces will bejy/. 
T his is eaiily demonftrated from the refolution of forces, 
2 . Through S draw SN parallel to PM or pm , & c ., and 
aflume in the line (SN) SP — FM and S p—pm, and let the 
force at P 7 in the line SN and diftance r^M'P 7 \ the force of the 
body moving in the curve at the diftance P'S : P'M' : SP' ; 
then if the velocity at the diftance SP = PM be P£, the velo- 
city at the diftance S p—pm will bey>/. 
Cor. The force in the direction of the line SL vanishes in 
the point where a perpendicular SN to the line SL parting 
through the point S cuts the curve, and confequently the 
velocity in the direction of SL in that point is the greateft or 
leaft, &c.; but if the tangent of the curve be perpendicular in any 
point to LS, then the velocity in the diredtion LS is nothing: 
the lame may be applied to the velocity in any other direction. 
Ex. Fig. i o. Let a body move in the circumference of a circle 
SPA, of which the center of force is a point S in the circum- 
ference ; it is known, that the force in the direction and at the 
diftance SP is as SP~5 ; but the force in the direction SP is by the 
hypothefis to the force in the direction (Sx^) :: SP : SM, if PM 
be perpendicular to SM, and confequently the force in the 
direction (SA) is as SM x SP~ 6 ; but if AS be a diameter, 
AS x SM = SP 2 ; therefore SM x SP~ 6 = SM x AS~ 3 x SM ~ 3 = 
- M ... ■ ; and the diameter AS being given, the force in the line 
AS 3 ° ° 
SA varies as SM - % that is, inverfely as the fquare of the dif- 
tance : if the force varies as SM~* then vv will vary as 
