Ye 



angle between the x axis and tlie projection of r upon the (xj) plane; b 

 being a constant. 



It will be assumed here that there is a force of friction equal and 

 opposite to tlie centrifugal force, of a particle (or wheel) moving down the 

 surface, under the action of gravity (g). If these equal and opposite vec- 

 tors be introduced, tlie problem reduces to that of determining the motion 

 of a particle (or wheel) on a fixed smooth surface. 



The general equation of kinetic energy* is, 



where m represents tlie mass, v the velocity and X, Y and Z the axial com- 

 ponents of the impressed forces. 



Denoting the angle between the [xy] plane and the tangent plane of 

 the surface by a there results : 



sin 2 a 



(2) X = mg sm a cos a cos u i:: mg — ~ — cosu. 



sin 2 a . 

 Y^mg sin a cos a sm u — mg „ — sin u. 



Z=mg. 

 And equation ( 1 ) reduces to 



, , , , r sin 2 a ., , sin 2 a . „ 1 , 



d ( Jmv2 ) = g — - — cos^ u + g — ^ sin2 u j m dr. 



r sin2a . , siii2a . , gb | , 



-|- — g — - — r sm u cos u + g — - — r sm u cos u -|- -^ j mdu ; or, 



r sin 2 a f , , mg b , 



(3) d(J mv2)=m!g— 2— J dr + ^-du. 



But the angle a equals, 



27rr 

 a = cos~^ — 7= — 



l/47r2r2 + b2 



„^ sin2a . 27j-rb , „ ,r,. 



Wlience — -„ — = sinacosa= . „ „ -r^r:i and from (3). 

 2 4 TT^ r^ -|- b'' 



, , „ r 2 TT b mg r 1 J , r mg b i , 



(4) d(^mv^):z. l47^YM?b^Jdr+ hVJd"- 



Tliis, upon integration, gives. 



eb 

 (5) v2 = |^log 



b^ 



' 4 71-2 



l^»~^47r2 J 



-}- — u, the initial conditions being v— 



and r^To when u^O. 



■• Ziwet Mechanics, p. 103, Vol. III. 

 t These are partial derivatives. 



