V' 



"1^2 Mr, ViNCE on the Motion of 



the motion vary as the fqnares of the velocities, hence 



2 2 



-^'. a (^.\ I : 2F) :: -T. ~ z : <2* - 2Fs = fquare of the progref- 



five velocity when the motion hecomes uniform ; therefore the 



velocity deflroyed by friction ~a — \/ a^ — 2F2 ; hence, as the 

 velocity generated or deflroyed in the fame time is in 

 proportion to the force, we have by Cor. 2, Prop. i. 



}• s : r a •.-.a — s/a" ~ 2bz \ - X a — s/a'' — z¥z the velocitv of 



r i "' 



tiie circumference efg generated about the center, conlequently 



s/a^ — 2Fz~ — X a — s/ii^ — 2b'z, and hence z— ^ 



r s as X 2F 



the Ipace which the body defcribes before the motion becomes 

 'uniform. 



2. If w^e fubftitutethis value of z into the expreffion for the 



velocity, we (liall have a x - for the velocity of the body when 



its motion becomes uniform ; hence therefore it appears-, that 

 the velocity of the body, when the fr](5lion ceafes, will be the 

 fame whatever be the quantity of the fri6lion. If the body be 

 the circumference of a circle, it will always lofe half the ve- 

 locity before its motion becomes uniform. 



Case II. i. Let the body, befides having a progreffive 

 velocity in the direction LM (fig. 3.) have alio a rotatory mo- 

 tion about its center in the diredlion gfe, and let v reprefent 

 the initial velocity of any point of the circumference about the 

 center, and fuppofe it firft to be lefs than a ; then fri«5lion being 

 a uniformly retarding force, no alteration of the velocity of 

 the point of contad of the body upon the plane can affecl the 

 ^quantity of fridion ; hence the progreffive velocity of the body 

 Hvill be the fame as before, and confequently the rotatqry velo-^ 



city 



