T6 



augle between the x axis and tlie projection of r upon tlie (xy) plane; b 

 being a constant. 



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

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

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

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

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



The general equation of kinetic energy* is, 



(1) d(Jmv2) = [xif-^+Y^+Z^l dr+ [x^ + Y^^ +Z^^1 du. 

 ^ - I dr dr ' dr J ' [ du ' du du J 



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 tlie tangent plane of 

 the surface by a there results : 



(2) X = mgsinacosacosu^mg — - — cosu. 



,^ . . sin 2 a . 



Y=mgsinacosasmu=^mg — - — smu. 



Z=mg. 

 And equation ( 1 ) reduces to 



, , , _ f sin 2 a „ , sin 2 a . „ 1 , 

 d ( ^mv^ ) = g — ^ — cos^ u + g — ^ — sm^ u m dr. 



, r sin2a . , sin2a . , gb] 



-f- — g ^ — r sm u cos u -f g — - — r sm u cos u -1-^ i mdu ; or, 



/ox J /] 9 X r sin2af , , mgb 



(3) d(J mv2) =m.g— g— J dr + — ^du. 



But the angle a equals, 



27rr 

 a = cos""^ —7=^ • 



-nm sin2a . 2Trb ^ „ .„. 



Wlience — -^ — = sm a cos a = -. — , , - . , o and from (.3) . 



This, upon integration, gives, 

 b^ 



■w*'£ _l_ 



(5) v^ = ||log 



^4 7r2 gb 



r— - -I- ^ u, the initial conditions being v =: 



[ ° ' 4-2 



and r^To when u = 0. 



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

 t These are partial derivatives. 



