VOL. LXXV.] PHILOSOPHICAL TRANSACTIONS. (j(J5 



5. If a — 0, or the body be placed on the plane without any progressive velo- 

 ., rs z x v z 



city, then z = ,. 



•" 2f x as 1 



Case hi. — 1. Let the given rotatory motion be in the direction gef\ then as 

 the friction must in this case always act in the direction ml, it must continually 

 tend to destroy both the progressive and rotatory motion. Now as the velocity 

 destroyed in the same time is in proportion to the retarding force, and the force 

 which retards the rotatory is to the force which retards the progressive velocity, 

 by Cor. 2, Prop. 1, as ra : rs, therefore if?; be to a as ra is to rs, then, the retard- 

 ing forces being in proportion to the velocities, both motions will be destroyed 

 together, and consequently the body, after describing a certain space, will rest ; 

 which space, being that described by the body uniformly retarded by the force f, 



will, from what was proved in case 1, be equal to — . 



1 2f 



2. If v bears a greater proportion to a than ra does to rs, it is manifest that 

 the rotatory motion will not be all destroyed when the progressive is ; conse- 

 quently the body, after it has described the space —, will return back, in the di- 

 rection ml ; for the progressive motion being then destroyed, and the rotatory 

 motion still continuing in the direction gef. will cause the body to return with an 

 accelerative velocity, till the friction ceases by the body's beginning to roll, after 

 which it will move on uniformly. Now to determine the space described before 



this happens, we have rs : ra :: a : — the rotatory veU city destroyed when the 



progressive is all lost ; hence v = -- = the rotatory velo- 

 city at that time, which being substituted for v in the last article of case 2, gives 



(v-rrs — a x r ay ^ ^ g g c j escr j|- >ec j before the motion becomes uniform. 

 2r x as 1 ' 

 3. If v has a less proportion to a than ra has to rs, it is manifest that the 

 rotatory motion will be destroyed before the progressive ; in which case a rotatory 

 motion will be generated in a contrary direction till the two motions become 

 equal, when the friction will instantly cease, and the body will then move on 

 uniformly. Now ra : rs :: v : the progressive velocity destroyed when the 



rotatory velocity ceases, hence a = — = progressive velo- 

 city when it begins its rotatory motion in a contrary direction ; substitute there- 

 fore this quantity for a in the expression for z in case 1, and we have 



(r,» + 2rs x ra) x (a x m - v x n)* for the described after the rotatory motion 



as 2 x ar 1 x 2f r 



ceases before the motion of the body becomes uniform. Now to determine the 

 space described before the rotatory motion was all destroyed, we have (as the 

 space from the end of a uniformly retarded motion varies as the square of the 



VOL. XV. 4 Q 



