VOL. LXXV.] PHILOSOPHICAL TRANSACTIONS. 663 



scent = ac/ ; now it is manifest, that the time of descent will COU- 

 TH x ra x bc 



tinue the same, if the friction be increased, for the body will still freely roll, as 

 no increase of the friction acting at a can affect the motion of the point s. 



If the body be projected from c with a velocity, and at the same time have a 

 rotatory motion, the time of descent and the number of revolutions may be de- 

 termined from the common principles of uniformly accelerated motions, as we 

 have already investigated the accelerative force of the body down the plane and 

 of its rotation about its axis; it seems therefore unnecessary to lengthen out this 

 paper with the investigations. 



Prop. 2. — Let the body be projected on an horizontal plane lm (fig. 3) with a 

 given velocity ; to determine the space through ivhich the body will move before it 

 stops, or before its motion becomes uniform. 



Case ]. — 1. Suppose the body to have no rotatory motion when it begins to 

 move; and let a = the velocity of projection per second measured in feet, and 

 let the retarding force of the friction of the body, measured by the velocity of 

 the body which it can destroy in one second of time, be determined by experi- 

 ment and called f, and let x be the space through which the body would move 

 by the time its motion was all destroyed when projected with the velocity a, and 

 retarded by a force p ; then, from the principles of uniformly retarded motion, 

 x = — ; and if t = time of describing that space, we have t = -; and hence 



.JF F 



the space described in the first second of time = — - — . Now it is manifest, 

 that when the rotatory motion of the body about its axis is equal to its progressive 

 motion, the point a will be carried backwards by the former motion as much as 

 it is carried forwards by the latter; consequently the point of contact of the body 

 with the plane will then have no motion in the direction of the plane, and hence 

 the friction will at that instant cease, and the body will continue to roll on uni- 

 formly without sliding with the velocity which it has at that point. Put there- 

 fore z = the space described from the commencement of the motion till it be- 

 comes uniform, then the body being uniformly retarded, the spaces from the 

 end of the motion vary as the squares of the velocities, hence 



— : a 2 (:: 1 : 2f) :: - z : a' 2 — 2sz = square of the progressive velocity when 



the motion becomes uniform; therefore the velocity destroyed by friction = a — 

 V a 2 — 2fz: hence, as the velocity generated or destroyed in the same time is in 

 proportion to the force, we have by cor. 2, prop. 1, rs : ra :: a — ^ a 2 — 2fz : 



-Xfo-^ ar — 2fz) the velocity of the circumference efg generated about the 



rs v 



centre, consequently V d 1 — 2fz = y X (a — V" a' 2 — 2fz), and hence z = 



