278 VII. DYNAMICS OF ROTATING BODIES. 



so that it appears that cp and ^ are cyclic coordinates. The forces 

 tending to change #, #, <p are 



p <W__df , , 



~dtd& d&~dt 



- {^' 2 sin#cos# C((p' 



P^ = J^==^{^sin 2 #V 



P *-TtW = Tt( C W+'1' 1 }' 

 If there is no force tending to change the spinning of the top, P 9 = 0, 



83) C O' + cos # ^') = const. = H,, 



which is the integral 71). Eliminating cp ! by means of this equation, 

 and forming the kinetic potential, 



TT 



84) (p' = -- 



85) ^=T-H 2 (p'= 1 -A(^' 2 



the second term containing ^' in the first power. Such terms in 

 the kinetic potential give rise to what have been called by Thomson 

 and Tait gyroscopic forces, whose theory has been treated in 50. 

 Using this form to determine the forces, we have 





The influence of the cyclic motion may be very simply shown if the 

 spinning body is mounted as a balanced gyroscope in gimbals, as in 

 Fig. 92. Suppose the vertical ring be held fixed. Then ^ = const., tjj' = 0, 



Spinning the inner ring about the horizontal axis requires the 

 same force P# whether the cyclic motion exists or not, whereas a 

 force is developed tending to make the vertical ring revolve about 

 its axis, which must be balanced by the force of the constraint, Py, 



proportional to -JT- On the other hand let us hold the inner gimbal 

 ring horizontal. Then # = ? &' = 0. 



