i8+ ^^^' ViNCE on the Motion of 



5. If a — 0, or the body be placed upon the plane wichoiit 



any progreffive velocity, then z — ~~~ . 



Case III. i. X.et the given rotatory motion be in tlie direc- 

 tion ^^y'; then as the friclion muft in this cafe alv/ays acl in 

 the direction ML, it miift continually tend to deftroy both the 

 progreflive and rotatory motion. Now as the velocity de- 

 flroyed 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 /(rijo-r^/i;^ velocity by Cor. 2. Prop. i. as r^:rj, 

 therefore if 1; be to ^ as ra is to r s^ then the retarding forces 

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

 flroyed together, and confequently the body, after defcribing 

 a certain fpace, will refl ; 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 -r. . 



2. If V bears a greater proportion to a than ra does to r s^ 

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

 when the progreflive is ; confequently the body, after it ha$ 



defcribed the fpace -r,, will return back in the diredlion ML; 



for the progreffive motion being then deftroyed, and the rota- 

 tory motion ftili continuing in the dire<£tion g efy will caufc 

 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 



Telocity deftroyed when the progreffive is all loft ; hencfi 



^ - '[fiif = Vi!2zzl^llf — the rotatory velocity at that time, whiclaL 



being 



