Centripetal Forces. 
Cor. From the forces given and the velocities in the above 
mentioned diredions at the point P, can be deduced the veloci- 
ties in the fame diredions at the point/, and confequently the 
tangent to the curve at the point p. 
prop. ix. 
1. Let the refinance of a body, moving in a right line, be as 
any fundion V of the velocity v ; then will i — ~ , x = 
; where /, v, and x , denote the increments of time, velo- 
city, and fpace ; their fluents properly correded will give the 
time and fpace in terms of the velocity. 
2. Let a body move in a right line, and be aded on by an 
accelerating force in that line, which varies as any fundion X 
of the diflance x from a given point ; and refilled by a force 
which is as any fundion V of the velocity ninto its denfity X 7 , 
which varies alfo as a fundion of x and v ; then will (X + tfVX 7 ) 
x— —vv, from its fluent x can be found in terms of v , or v 
in terms of x; and thence / = 
of which the fluent 
X+tfVX 7 
properly correded gives the time. 
Ex. i. Let V =v 2 and X / a fundion of x; that is, let the 
refinance be as the fquare of the velocity and denfity, whence 
(X + ^‘P 2 X / )^= — vv, of which equation the fluentiai will be 
e J iaXx __ _ r ' e fiaXx ^ X* + A, and t — 
2 J 
— h B, where A and B are 
Sv - (<? l 2flX X (/ J 2aX * x Xx+A)) 
invariable quantities to be afiumed according to the condi- 
tions of the problem,. 
