Bodies ajj'edled hy Friction* 1 85 
being fubftituted for v in the laft article of Cafe II. gives 
z 
for the fpace defcribed before the motion becomes 
2i* Xas 1 
uniform* 
3. If v has a lefs proportion to a than r a has to r s , it is 
manifest, that the rotatory motion will be deftroyed before the 
progrejjive ; in which cafe a rotatory motion will be generated 
in a contrary direction until the two motions become equal, 
when the friction will inftantly ceafe, and the body will then 
‘ move on uniformly. Now r a : r s :: v : V ~~ the progreffive 
velocity deftroyed when the rotatory velocity ceafes, hence 
a ~~Tir~ ra “ progreliive velocity when it begins its 
rotatory motion in a contrary direction ; fubftitute therefore 
this quantity for a in the expreftion for % in Cafe I. and we have 
— * - . — - 
rs z 2rs x ra X a X ra—v Xrs r i r . r .. , , 
2 2——^ ror the Ipace delcnbed atter the rota- 
as X ar X 2? r 
tory motion ceafes before the motion of the body becomes 
uniform. Now to determine the fpace defcribed before the 
rotatory motion was all deftroyed, we have (as the fpace from 
the end of a uniformly retarded motion varies as the fquare of 
the velocity) a z : 
a a X ra-r-v Xrs 
• • 
2 F * * 
ra 
aXra — v xrs 
2F X ra" 
the fpace that 
could have been defcribed from the time that the rotatory 
velocity was deftroyed, until the progreffive motion would 
have been deftroyed had the frifton continued to aft ; hence 
« aXra — v X rs 
2 F 2 F x ra 
lav X raX rs — v z x rs z 
2F x ra 2 
= the fpace defcribed when 
the rotatory motion was all deftroyed, hence 
rs 2 + 2 r s X ra X a X ar — v x ' r 
as 2 X ar 2 x 2 F 
+ 
lav X ra X rs — v 1 x rs z 
2 F x rd~ 
whole fpace de- 
fcribed by the body before its motion becomes uniform. 
Vol* LXXV. B b 
o 3- 
