Bodies affected by Friction* 1 85 



being fubflituted for v in the lail article of Cafe II. gives 



'V -f rs ~a x:a 



for the fpace defcribed before the motion becomes 



2F AUi^ 



uniform. ' 



3. If "0 has a iefs proportion to a than ?- a has to r s, it is 

 manlfefl:, 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 fridion will inftantly ceafe, and the body will then 



move on uniformly. Now r a : r s :: v : '—^ the progreffivc 



velocity deftroyed when the rotatory velocity ceafes, hence 



a — — = =:progreiiive velocity when it begins its 



rotatory motion in a contrary diredion ; fubflitute therefore 

 this quantity for a In the expreffion for % in Cafe L and we have 



n -{■ 2>'! A 'a X a X ra — v X TS ^ t r ^ r •^ i r i 



2 2 — -^ tor the Ipace deicribed after the rota- 



cii X at X 2r ^ 



tory motion ceafes before the motion of the body becomes 

 unifjrm. 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 



^t 1 V \ a ^ aX^a — vXrs a Xra — v Xrs , -- , 



the velocity ) a : -^ : : ^ : v. z — the fpace that 



•J y 2F ra 21" X ra ^ 



could have been defcribed from the time that the rotatory 

 velocity was deftroyed, until the progreffive motion would 

 have been deftroyed had the fri(Slon continued to a (ft ; hence 



fi aX ra — v Xrs lav X ra X rs — 7;" X rs^ 



2F 2F X ra" ~ W777 == ^^^^ ^P^^^ defcribed when 



the rotatory motion was all deftroyed, hence 



♦ ,1 



.1 



fi +2rsxiaxaxa —vx 2avXraXrs — vxrs 



as^xur^x2V + ^ ^F777 = whole fpace de- 



fcribed by the body before its motion becomes uniform. 



Vol. LXXV. B b d f.- 



