6Q4 PHILOSOPHICAL TRANSACTIONS. [ANNO 1785. 



(rs" 1 + 2rs x ra) x a tne spa ce which the body describes before the motion becomes 



as* x 2f r ' 



uniform. 



2. If we substitute this value of z into the expression for the velocity, we shall 



have a X — for the velocity of the body when its motion becomes uniform; hence 



rs J 



therefore it appears that the velocity of the body, when the friction ceases, will 

 be the same, whatever be the quantity of the friction. If the body be the cir- 

 cumference of a circle, it will always lose half the velocity before its motion be- 

 comes uniform. 



Case 2. — 1. Let the body, besides having a progressive velocity in the direc- 

 tion lm (fig. 3) have also a rotatory motion about its centre in the direction gfe 

 and let v represent the initial velocity of any point of the circumference about 

 the centre, and suppose it first to be less than a; then friction being a uniformly 

 retarding force, no alteration of the velocity of the point of contact of the body 

 on the plane can affect the quantity of friction; hence the progressive velocity of 

 the body will be the same as before, and consequently the rotatory velocity gene- 

 rated by friction will also be the same, to which if we add the velocity about the 

 centre at the beginning of the motion, we shall have the whole rotatory motion ; 



hence therefore, v -f- — X (a 2 — ^ a 1 — 2fz) = V Q - — 2fz, consequently z = 



a- x as — _( t> x >M -_a_ x >a) tne space d escr j D ed before the motion becomes uni- 



2f X as 1 r 



form. 



2. If this value of z be substituted into the expression for the velocity, we 



shall have — — — ?-?— for the velocity when the friction ceases. 



3. If v = a, then z = O, and the body will continue to move uniformly with 

 the first velocity. 



4. If v be greater than a, then the rotatory motion of the point a on the plane 

 being greater than its progressive motion, and in a contrary direction, the abso- 

 lute motion of the point a on the plane will be in the direction ml, and conse- 

 quently friction will now act in the direction lm in which the body moves, and 

 therefore will accelerate the progressive and retard the rotatory motion ; hence it 

 appears, that the progressive motion of a body may be accelerated by friction. 

 Now to determine the space described before the motion becomes uniform, we 

 may observe, that as the progressive motion of the body is now accelerated, the 

 velocity after it has described any space z will be =. V a 1 + 2fz, hence the velo- 

 city acquired = ^ a 1 -f 2fz — a, and consequently the rotatory velocity de- 

 stroyed y s X (^a 2 + 2fz — o), hence v — — X (Va 5r +"2Fz — a)=^a 2 +2pz7 



,l r ('•* X r + ra x a) 2 — a a x as- , . , 



therefore z — ^ — r ' — the space required. 



