67] CYCLIC SYSTEMS. 131 



*- = \4 (*' + *' cos 0) cos 



+ A (<j>' + y cos 0) sin - .Bip sin d cos 0, 



If there is no force tending to change the rotation of the gyrostat 

 in its ring 



P*=0, 4(f + ^r'cos0) = c, 

 and eliminating <fi by means of this equation, 



f -J'-'f cosft 

 <1> = T-c0' = -|^ + ^ (<9' 2 + i|r' 2 sin 2 0) + c^' cos 0. 



the last term containing ^r' in the first power. Using this form of 

 4> to determine the forces, we obtain 



' - ^' 2 sin ^ cos + of sin A 



The influence of the cyclic motion may be most simply shown 

 if the vertical ring be held fixed. Then -fy = const., and i/r 7 = 0, 



= c sm V -r- , 

 at 



Spinning the inner ring about the horizontal axis requires the 

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

 is developed tending to make the vertical ring revolve about its 



rlf) 

 axis, which must be balanced by the force c sin 6 -j- . This 



force at once shows that there is a concealed motion, even if the 

 disposition of the concealed parts be unknown. This is exem- 

 plified in the gyroscopic pendulum, which is simply a pendulum 

 with two degrees of freedom, containing a gyrostat whose axis is 



92 



