1 88 Mr,yn;cTS. on the Motion of 



hence if we put z— PC, /) = the circumference of a circle 

 whole radius is unity, then will pz = circumference defcribed 



by the point P ; confequently — ^=: the number of revolutions 



required. 



Cor. It the folid be a cyllnJer and r be the radius of its bafe, 



then 2, = — , and therefore the number of revolutions 



4 ' "" X.W...WVX V.X .V..V.XV..XV...O-^^^p. 



PROPOSITION V. 



91? ^*f«£/ //6<? nature of the curve defcribed by any point of a body 

 qfcdUd by friBion, when it defends down any inclined plane. 



Let efg (fig. 5.) be the body, the points ^, r, j, as in Prop. L 

 and conceive j /, r ;/, to be two indefinitely fmall fpaces defcribed 

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

 will reprelent the velocities of thofe points ; but from Prop. I. 

 the ratio of thefe velocities is cxprefled by m x CB : a x CA, 

 hence s t : r n i: m x CB : a x CA. With the center r let a 

 circle 1; w be defcribed touching the plane LM 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 Aiding. To determine which radius^ 

 produce r s to ;v, parallel to which draw n dy, and produce n t 

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

 ;4bove-mentioned, the velocity of the point x mufl be to the 

 velocity of the point r as 2; :.i,. that is, %x : y x :: 2 : 1^ 

 hence zyz:zyx = nr. Now zy : dt {\: ny : nd) n rx i r si 



therefore ^/=:— x%y=:— x «r, hence ^j (=:td+ds = fd-^nrzz 

 4 ^^ 



