TOL. XXVI.J PHILOSOPHICAL TRANSACTIONS. 441 



Let now a body move in the curve qao, fig. 1, by means of a centripetal 

 force tending to s ; and let the velocity of the body at a be called c , also the 

 velocity with which a body, urged by the same force, at the same distance, can 

 move in a circle, be called c. It appears from the first theorem, that if sa ex- 

 press the centripetal force tending to s ; then the centripetal force tending to r, 

 by means of which the body, with the velocity c, may describe a circle whose 

 radius is ar, will be expressed by sp. But the centripetal forces of bodies de- 

 scribing circles, are as the squares of the velocities applied to the radii of the 



circles. Therefore it will be, sp : sa :: — : — ; hence sf.ar : sa* :: c^ : c% and 



c : c :: i/ sp.ar : sa. 



If SP coincide with sa, as is the case in the vertices of the figures, it will be 

 c : c :: i/ ar : y' sa. So that if the curve be a conic section, ar the radius of 

 curvature at its vertex, is equal to half the latus rectum, or ^l ; and therefore 

 the velocity of the body in the vertex of the section, will be to the velocity of 

 the body describing a circle at the same distance, in the subduplicate ratio of 

 the latus rectum to the double distance. 



0'„ SA.5A .1 o ■» Sr.SA.5A , sr.SA 



Smce AR = — —^ then c^: c^:: — -: — : sa* :: —r- : sa :: sp.sa : sa.sp. 



SP SP SP 



Therefore, from the given relation of sp to sa, the ratio of c to c will be given. 

 For example, if the force be reciprocally as the m power of the distance, that 



is, if —, — = -r — ; then sp — — ; therefore c^ : c' :: sp.sa : — :: 



"' SPJ.SA a*SA"» fl*SA"« A*SA„ 



a^sA"*"' : Zjsp*. Whence if we put sp' = — ^ — = — — Ir— — > »t will be c*: 

 c^ :: a^SA"*"' : m — 1.4-a^SA'"-' :: m — 1 : 2 ; and therefore c : c :: ^^2 ; 



y^ m — 1. 



But if sp* = , = — , — ■ — , then c : c :: a'^sA"' ' : — . „ , — , 



that is, as i — esA*"*"' to w -- l.-l-b. But the ratio of b — eSA"*"' to m — 1.^^, 

 is less than the ratio of ^ to w — I .-i^i, or than the ratio of 2 to m — I ; hence 

 c will be to c, in a less ratio than v^2 to ^ m — 1. 



In like manner, if there be taken sp = ; ,, it will be found that c 



i + csA*"-*' 



will be to c, in a greater ratio than that of -/ 2 to \^ m — 1 . 



Corol. — If a body move in a parabola, and the centripetal force tend to the 

 focus s ; then the velocity of the body, will be to the velocity of a body de- 

 scribing a circle at the same distance, every where as V' 2 to 1 : for in that case 



against Mr. Keill to the Royal Society, which occasioned the famous dispute concerning the invention 

 of fluxions. 



VOL. V. 3 L 



