Mr. Vince on the Motion of 
5. If a~o, or the body be placed upon the plane without 
o - ys ^ X * 0 ^ 
any progreffive velocity, then, z - 2 i r~~ • 
Case III. 1. Let the given rotatory motion be in the direc- 
tion g e f\ then as the fridion muft in this cafe always ad in 
the diredion ML, it muft continually tend to deftroy both the 
progreffive and rotatory motion. Now as the velocity de- 
ftroyed in the fame time is in proportion to the retarding force, 
find the force which retards the rotatory is to the force which 
retards the progreffive velocity by Cor. 2. Prop, 1. as ra : r 
therefore if v be to a as ra is to rs, then the retarding forces 
being in proportion to the velocities, both motions will be de- 
ftroyed together, and confequently the body, after describing 
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 r /, 
it is manifeft, that the rotatory motion will not be all deftroyed 
when the progreffive is ; confequently the body, after it has 
defcribed the fpace -p , will return back in the diredion ML ; 
for the progreffive motion being then deftroyed, and the rota- 
tory motion ftill continuing in the diredion g ef 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 deftroyed when the progreffive is all loft ; hence 
v _ raXa — vx rs ~~ ax . r J — the rotatory velocity at that time, which 
