t)n the Motion of Bodies affected ly Friction. 1 1 9 



ff'KOPOSITIO^ V. 



To find the nature of the c?irv£ descriled hg anv point of 

 a I'odn affected by friction, when it descends down any in- 

 clined plane. 



Let efg (fig. 5.) be the body, the pohits a, r, s, as iu 

 Prop. /.. and caiiccive st, rn, to be two indelinitely small 

 spaces described by tlte points *■ audr in the *arme time, aitd 

 which therefore unll represent t-kr velocities of those points; 

 but from Prop, L the ratio of these velocities is expressed 

 by m X CB : a x CA, hence s.t : r n : : m x CB :: e x 

 CA^ With the center;" let a circle mw be described toach- 

 ing tlie plane LM \v.hlch is parallel to AC at the point b, 

 ,and let the radius of this circle be such th?,t, concei\'ing it 

 to descend upon tiie plan« LM -along with the body de- 

 scending on CA, the point h may be at rest, or the circle 

 may roll without sliding. To determine which radius, pro- 

 duce r.s to .r, parallel to which draw ndy, and produce nt 

 to z ; now it is manifest, that in order to answer the con- 

 •ditions above mentioned, the velocity of the point x must 

 be to the velocity of the point r as 2 : 1, that is, zx : yx 

 :: 2 : 1, \i<iQ.<x, z y = y x = n r^ Now xy : dt (:: ny : 



nd) -.x rx : rs; therefbr-c £?f = — Kzy=— xnr, hence 



rx ^ rx 



t s (= td + ds ■=: id ri-nr = — x nr+nr) = — -_ — xnr, 



r,i' Tj, 



consequently ■ : \ :: t s -. nr ': (from \\ hat is proved 



above) m x CB .: a x CA ; therefore a x CA X rs -\- a x 



r^ « g-\n -L. o, X CA x rs 



CA X rx = mx CB x rx, hence rx = 



w X CB — a X CA 



the radius of the circle which rolling down the inclined 

 plane LM, and carrying the body with it, will give the 

 irne ratio of its progrcssiv.e to its rotatory motion, and con- 

 sequently that point of the circle which coincides with any 

 given point of Llie .l)ody will, as the circle revolves upon the 

 line LMi describe the same c.unve as the corres|)onding 

 point (if the body ; but as the nature of the curve described 

 by any point of a circle revolving upon a straight line is al- 

 re<idy very well known^ it >ce;us unijecess.ary to give die 

 investigation. 



By a method of reasoning, not verv diflerent, may the 

 nature of the curve, which is described by any point of a 

 body moving upon an horizontal plane, and affected by 

 friction, be deternitned. 



H 4 XV' . Cop]/ 



