On the Motion of Bodies affected ly Friction. 1 1 7 



scribed before the rotatory motion was all destroyed, \\ e 

 have (as the space from the end of a uniformly retarded 



motion varies as the square of the velocity "1 a- : — :: 



2"^ 



a X ra— V X rs a x ra — -r x >'i , , , 



■ ; : — - „ the space that could 



r a- 2 1' X ■'" a - 



have been described from the time that the rotatory velocity 



was destroyed, until the progressive motion would have 



been destroyed had the friction continued to act ; hence 



a- a X r a — u x r s 2 a v x ra x r s — i' - x rs- . 



2^F 2F x'r'a'- "~ 2 F x r a- 



space described when the rotatory motion was all destroyed. 



, Ts- -Y^rsxra xaxar — vxsr '2gv xraxrs—v' 

 hence ■ , — - - - , „ i- 



as- X ar- x 2F iF x r a- 



= the whole space described by the body before its motion 

 becomes uniform. 



DEFINITION. 



The CENTER of FRICTION is that point ifi the lase of a 

 lady on which it revolves, info which if the whole surface 

 of' the base, and the mass of the body were collected, and 

 made to revolve about the center of the base of t lie ^iven body, 

 ihfi angular v/docity destroi/ed by its friction would be eq^ial 

 to the angular velocity destroyed in the given body by its 

 J'riction in the same time. 



PROPOSITION III. 



To find the center of friction. 



Let FGH (fig. 4.) be the base of a body revolving about 

 its center C', and suppose about a, I, c, &c. to be indefi- 

 nitely small parts of the base, and let A, B, C, Sec. be 

 the corresponding parts of the solid, or the prismatic parts 

 havinir a, b, c, hic. for their bases; and P the center of 

 friction. Now it is manifest that the decrement of the an- 

 gular velocity must vary as the whole diminution of the 

 momentum of rotation caused by the friction directly, and 

 as the whole momentum of rotation or effect of the inertia 

 of all the particles of the solid inverscdu ; the former being 

 employed in diminisb.intr the angular vcloeitv, and the latter 

 in opposing that diminution by the endeavour of the par- 

 ticles to persevere in their motion. Hence, if the effect of 

 the frictioti varies as the efl'ect of the inertia, the decre- 

 ments of the angular velocity in a given time will be equal. 

 Now as the (juanlity of friction (as has been proved from 

 experiments) doci: not depend on the velocitv, the efl'ect of 



