Chessin — On the Motion of Gyroscopes. 33 



mined by the condition sin 6q = 0, i. e. ^ = or tt. Hence, 

 the axis of the torus can be in equilibrium only if it is parallel 

 to the axis of the earth. 



Indeed, this fact can be readily ascertained by the ordinary 

 rules of mechanics. Moreover, it is clear, that one of the two 

 positions 6^ = or tt, in which the positive axis of the torus is 

 parallel to the positive axis of the earth, is the position of stable 

 equilibrium J while the other position is that of unstable equilib- 

 rium; i. e. if i/t'q > 0, then ^^ = tt is the position of stable 

 equilibrium and ^o = ^ the position of unstable equilibrium, 

 while for y^\ < the position ^^ = is that of stable equilib- 

 rium and 6q ■= 77 one of unstable equilibrium. 



The motion is completely determined by the equation (26) 

 and the following formulas giving the angles (f> and -^jr. 



I 



^ = cf>, + o>t+ I ,; ■ A .;»2 a dt, 



t 



I, 



d-{-b sin^ 







t=^.+v- \ -^-^i:s^fr'' '^- 



It should be observed that 



cos 



28 2&) / 



^^_cos77= ^ = ^ J(cZ + 6sin»^)[A(6 + d) — 6Z32] 



i. e. the difference cos rj, — cost] is a quantity of the order 

 of -^ • The angular velocity of rotation of the earth (w) 



being very small and ^\ by hypothesis, very great, the oscil- 

 lations of the axis of the torus (and of the ring ( C^) with it) 

 will be exceedingly small. The same is true with regard to 



