190 MOTION UNDER CONSTRAINTS AND RESISTANCES. [CHAP. X. 



from general principles sufficient integrals of the equations of 

 motion to determine the velocity completely. 



Let the axis of revolution be the axis x (x being measured 

 upwards), and let the particle at time t be at distance y from the 

 axis, and be on a meridian curve of the surface in an axial plane 

 making an angle < with a given axial plane, and let a- be the arc 

 of the meridian from some particular circular section to the 

 position of the particle. 



Then it is clear that the velocity along the tangent to the 

 meridian is cr, and the velocity along the tangent to the circular 

 section is y<j>. Thus the energy equation is 



= const. 



Fig. 52. 



Again, since the reaction of the surface on the particle is along 

 the normal to the surface, and the normal meets the axis of 

 revolution, while the weight of the particle acts in a line parallel 

 to this axis, the forces acting on the particle have no moment 

 about this axis. 



Hence by the Principle of the Conservation of Moment of 

 Momentum the moment of the velocity about the axis is constant, 

 or we have 



y z <j> = const. 



The equations written down determine cr and <, that is they 



