254 



VII. DYNAMICS OF ROTATING BODIES. 



The angle turned by the body in rolling the arc ds 2 on its equal 

 ds 1 is dr 2 dr ly and the angular velocity 



> (d fl ^ ( ~\ 



/1 1\ 



I 



e s Pj/ 



Fig. 72 c. 



Now if w denote the angular velocity with which the instantaneous 

 axis is turning about an axis through perpendicular to the common 

 tangent plane to the cones, we have 



ds 



which inserted in 8) gives 



9) 



Q l - Q. 



If the cones have external contact, 

 $ 2 is negative, and if we consider the 

 absolute values, we must take the 

 sign plus. 



Fig. 73. 



or\ 



8') o? = 



9 ') ^ 





Consequently if we give the values at every instant of three of 

 the quantities, co, the angular velocity about the instantaneous axis, 

 w, the angular velocity of change of the instantaneous axis, and ^ 



