VOL. LXXV.] PHILOSOPHICAL TRANSACTIONS. U(J7 



centre of its base, to determine the number of revolutions which the body ivill make 

 before all its motion be destroyed. 



Let the friction, expressed by the velocity which it is able to destroy in the 

 body if it were projected in a right line horizontally in one second, be determined 

 by experiment, and called f ; and suppose the initial velocity of the centre of 

 friction p about c to be a. Then conceiving the whole surface of the base and 

 mass of the body to be collected into the point p, and (as has been proved 



a 1 

 in prop. 2) — will be the space which the body so concentrated will describe be- 

 fore all its motion be destroyed ; hence if we put z = pc, p = the circumference 

 of a circle whose radius is unity, then will pz = circumference described by the 



point p ; consequently -— = the number of revolutions required. 



Cor. If the solid be a cylinder, and r be the radius of its base, then z = \r, 



2a 2 



and therefore the number of revolutions = - — . 



Sprv 



Prop. v. — To find the nature of the curve described by any point of a body 

 affected by friction, when it descends down any inclined plane. 



Let efg (fig. 5) be the body, the points a, r, s, as in prop. 1, and conceive st, 

 rn, to be two indefinitely small spaces described by the points s and r in the same 

 time, and which therefore will represent the velocities of those points ; but from 

 prop. 1, the ratio of these velocities is expressed by m X cb : a X ca, hence st : 

 rn :: m X cb : a X ca. With the centre r let a circle vw be described touching 

 the plane lm which is parallel to ac at the point b, and let the radius of this 

 circle be such that, conceiving it to descend on the plane lm along with the body 

 descending on ca, the point b may be at rest, or the circle may roll without 

 sliding. To determine which radius, produce rs to x, parallel to which draw ndy, 

 and produce nt to z ; now it is manifest, that in order to answer the conditions 

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

 ras2: 1, that is, zx : yx :: 1 : 1, hence zy = yx = nr. Now zy : dt (:: ny : nd) 

 :: rx : rs ; therefore dt = — X zy = — X nr, hence ts (= td + ds = id -f nr = 



— X nr 4- nr) = '* '* X nr, consequently : 1 :: ts : nr :: (from what is 



rx ' rx rx 



proved above) m X cb : a X ca , therefore a X ca X rs + a X ca X rx = m 



X cb x xr, hence rx = the radius of the circle which, rolling: 



^ ~ > Wi x cb — a X CA ° 



down the inclined plane lm, and carrying the body with it, will give the true 

 ratio of its progressive to its rotatory motion, and consequently that point of the 

 circle which coincides with any given point of the body will, as the circle revolves 

 on the line lm, describe the same curve as the corresponding point of the body ; 

 but as the nature of the curve described by any point of a circle revolving on a 



4 a 1 



