184 ^^' ViNCE on the Motion of 



5. If a — o, or the body be placed upon the plane without 



any progreflive velocity, then z = ^ — ^,<. 



Case III. i. Let the given rotatory motion be in the direc- 

 tion ^^y^; then as the frifbion mufl in this cafe always ad in 

 the dire£lion ML, it mufl continually tend to deftroy both the 

 progreflive and rotatory motion. Now as the velocity de- 

 ftroyed in the fame time is in proportion to the retarding force, 

 and the force which retards the rotatory is to the force which 

 retards the progrejive velocity by Cor. 2. Prop. 1. as r^ : r J, 

 therefore if 1; be to ^ as r^ is to r j, then the retarding forces 

 being in proportion to the velocities, both motions will be de- 

 ftroyed together, and confequently the body, after defcribing 

 a certain fpace, will reft ; which fpace, being that defcribed 

 by the body uniformly retarded by the force F, will, from 



what was proved in Cafe I. be equal to — , . 



2. If v bears a greater proportion to a than ra does to f s^ 

 it is manifeft, that the rotatory motion will not be all deftroyed 

 when the progreflive is ; confequently the body, after it has" 



defcribed the fpace — , will return back in the dIre£lIon ML; 



for the progreflive motion being then defl:royed, and the rota- 

 tory motion fl:Ill continuing in the diredtion gef^ will caufe 

 the body to return with an accelerative velocity until the fric- 

 tion ceafes by the body's beginning to roll, after which it will 

 move on uniformly. Now to determine the fpace defcribed 



before this happens, we have r s : r a :: a : — — - the rotatory 



velocity defl:royed when the progreflive is all lofl: ; hence 



rax a vxrs — axra ^i ^ ^ « • i ^ ^' U" U 



-y = _ — the rotatory velocity at that tmie, whicn 



/ being 



