6(J2 PHILOSOPHICAL TRANSACTIONS. [ANNO 1/85. 



then, because no force acting at a can affect the motion of the point s, that 

 point, notwithstanding the action of the friction at a, will always have a motion 

 parallel to ca uniformly accelerated by a force equal to that with which the body 

 would be accelerated if it had no friction ; hence, if 1m = Zl\ feet, the velocity 

 acquired by the point s in the first second will be = ; now the excess of 



the velocity of the point s above that of r, the centre, is manifestly the velocity 



with which s is carried about r; hence the velocity of s about the centre = 



2>« x cb _ 2t»xcb — Ca x ca ,, 2m x cb — 2a x ca 

 2a = , consequently rs : ra :: ■ — : 



CA CA ^ ' CA 



2w x ra x cb - 'a x ra x ca _ j-jjg velocity with which a point of the circumference 

 is carried about the centre, and which therefore expresses the force which acce- 

 lerates the rotation ; now as la expresses the accelerative force of the body down 

 the plane, and the spaces described in the same time are in proportion to those 



r , 2>« x ra x cb — 2a x ra x ca m x ra x cb — a x ra x ca 



forces, we nave 2a : ca :: : 



rs x ca ax rs 



the space which any point of the circumference describes about the centre in the 

 whole time of the body's descent down ca; which being divided by the circum- 

 ference/) X ra (where p = 6.283 &c.) will give '" x ~ — — : — for the whole 



number of revolutions required. 



Cor. 1. If a X ca = m X bc, the number of revolutions = O, and therefore 

 the body will then only slide; consequently the friction vanishes. 



Cor. 2. Let a'r's (fig. 2) be the next position of ars, and draw tr'b parallel to 

 sa; then will s't represent the retardation of the centre r arising from frictiqn, 

 and a'b will represent the acceleration of a point of the circumference about its 

 centre; hence the retardation of the centre: acceleration of the circumference 

 about the centre :: s't : a'b :: (by sim. As) tr : br' :: rs : ra. 



Cor. 3. If a' coincides with a, the body does not slide but only roll; now in 

 this case ss' : rr :: as : ar; but as ss and rr represent the ratio of the velocities 



of the points s and r, they will be to each other as : la or as m X cb : a 



X CA; hence, when the body rolls without sliding, as : ar :: in X cb : a X ca. 



Cor. 4. The time of descent down ca is = y/ — ; but by the last cor. when 

 the body rolls without sliding, a = 

 HUHfL hence the time of descent in that case = ac</ ; now the 



sa x ac m x ra x bc 



AC 



time of descent, if there were no friction, would be = —. , hence the 



v m x bc 



time of descent, when the body rolls without sliding: time ot' free descent :: 

 a/ sa: V ra. 



Cor. 5. By the last cor. it appears, that when the body just rolls without slid- 

 ing, or when the friction is just equal to the accelerative force, the time of de- 



