•i88 'Mr. ViNCE on the Motion of 



hence if we put a™ PC, j!? = the circumference of a circle 

 whofe radius is unity, then will /;s = circumference defcribed 



by the point P ; confeqiiently "^= the number of revolutions 



required. 



Cor. Jf the folid be a cylinder and r be the radius of its bafe, 



then z = ~ , and therefore the number of revolutions = 



4' Ipr^' 



PROPOSITION V. 



To find the nature of the curve defcribed by mzy point of a body 

 Ctfecicd by fri5lion^ when it defends do'wn ary inclined plane. 



Let efg (fig. 5 ) be the body, the points a, r, s, as in Prop. I, 

 and conceive st,r n, to be two indefinitely fmall fpaces defcribed 

 by the points s and r in the fame time, and which therefore 

 will reprefent the velocities of thole points ; but from Prop. I. 

 the ratio of thefe velocities U exprefled by m x CB : a x CA, 

 hence st^: rn :: m x CB : a x CA. With the center r let a 

 circle i; li' be defcribed touching the plane JLM which is parallel 

 to AC at the point b, and let the radius of this circle be fuch 

 that, conceiving it to defcend upon the plane LM along with 

 the body defcending on CA, the point b may be at reft, or the 

 circle may roll without Hiding. To determine which radius, 

 produce r s to .y, parallel to which draw n dy, and produce n t 

 to % ; now it is manifefl, that in order to anfwer the conditions 

 above-mentioned, the velocity of the point x muft be to the 

 velocity of the point r as 2:1, that is, %x ',yx\\ z\ i, 

 hence %yz=.yx = nr. Now zy : dt (:: ny \nd} \\ rxws'i 



therefore dt=L- x%yzz— x^»r, hence ^i (^^td-^dszzid-^-nrzz 

 4 rs 



