1 16 On the Motion of Bodies affected hj Priction^ 



then the retarding forces being in proportion to the veloci^ 

 ties, both motions will be destroved together, and conse- 

 quently the body, after describinaa certain space, will rest > 

 vliich space, being that described by the body uniformly 

 retarded by the force F, will, from what was proved in 



Case I. be equal to - ■ . 



2. If » bears a greater proportion to a than ra does to 

 rs, it is manifest that the rotatory motion will not be all 

 destroyed when the progressive is ; consequently the body, 



after it has described the space — , will rclurn back in the 



direction ML ; for the progressive motion being then de- 

 stroyed, and the rotatory motion still continuing in the 

 direction g e f, will cause the body to return with an ac- 

 celerative velocity until the friction ceases by the body's 

 bc2;inning to roll, after which it will move on uniformly. 

 Now to determine the space described before this happens, 



we havers \ ra :: a : — the rotatory velocity dc- 



rs 



stroved when the progressive is all lost ; hence v — — = 



^ — ! = the rotatory velochy at that time, which 



being substituted for v in the last article of Case II. gives 



—— — for the space described before the motion 



2F X as'- 



becomes uniform. 



3. If V has a less proportion to a than r a has to rs, it 

 is manifest that the roiatonj motion will be destroyed before 

 the progressive ; in which case a rotatory motion will bu 

 venerated in a contrary direction until the two motions be- 

 come equal, when the friction will instantly cease, and the 

 body will then move on uniformly. Now ?■« : rs :: v : 



the progressive vclocify destroyed when the rotatory 



V y. rs 



r a 



, . , V x r s a X ra — v x r s 



velocity ceases, hence a = = pro- 



ra r a 



o-ressive velocity when it begins its rotatory motion 

 m a contrary direction : substitute therefore this quan- 

 tity for a in the expression for z in Case I. and we have 



rs- + 2 rs X ra X (I X r a — v x r s ,, . , •. , 



. .^...- for the space described 



us- X ar- X 21' 



after the rotatoiy motion ceases before the motion of the 

 body becomes uniform. Now to cletcmaine the space de- 

 scribed 



