279-280] GYROSTATIC SYSTEM WITH FRICTION. 507 



There is no difficulty in shewing, with the help of criteria given by Routh*, 

 that if, as in our case, the quantities 



are all positive, the necessary and sufficient conditions that this biquadratic 

 should have the real parts of its roots all negative are that c x , c 2 should both 

 be positive. 



If we neglect terms of the second order in the frictional coefficients, the 

 same conclusion may be attained more directly as follows. On this hypothesis 

 the roots of (ii) are, approximately, 



X= -a 1 icr 1 , -a 2 iV 2 ........................ (iii), 



where o- l5 (T 2 are, to the first order, the same as in the case of no friction, viz. 

 they are the roots of 



a 1 a 2 (r 4 -(a 2 c 1 + a 1 c 2 +/3 2 ) o- 2 + c 1 c 2 = .................. (iv), 



whilst a x , a 2 are determined by 



.(v). 



It is evident that, if o-j and o- 2 are to be real, c lt c 2 must have the same sign, 

 and that if a lt a 2 are to be positive, this sign must be +. Conversely, if c l9 c 2 

 are both positive, the values of o^ 2 , cr 2 2 are real and positive, and the quantities 

 c i/ a i) G 2/ a 2 both h" 6 m tne interval between them. It then easily follows from 

 (v) that a l5 a 2 are both positivet. 



If one of the coefficients c lt c 2 (say c 2 ) be zero, one of the values of <r (say 

 o- 2 ) is zero, indicating a free mode of infinitely long period. We then have 



(vi), 

 (vii). 



* Advanced Rigid Dynamics, Art. 287. 



t A simple example of the above theory is supplied by the case of a particle in 

 an ellipsoidal bowl rotating about a principal axis, which is vertical. If the bowl 

 be frictionless, the equilibrium of the particle when at the lowest point will be stable 

 unless the period of the rotation lie between the periods of the two fundamental 

 modes of oscillation (one in each principal plane) of the particle when the bowl is at 

 rest. But if there be friction of motion between the particle and the bowl, there will 

 be ' secular ' stability only so long as the speed of the rotation is less than that of the 

 slower of the two modes referred to. If the rotation be more rapid, the particle 

 will gradually work its way outwards into a position of relative equilibrium in which 

 it rotates with the bowl like the bob of a conical pendulum. In this state the system 

 made up of the particle and the bowl has leas energy for the same angular momentum 

 than when the particle was at the bottom. Cf. Art. 235. 



