290 V 11 - DYNAMICS OF ROTATING BODIES. 



equations for the small vibrations of a system of two degrees of 

 freedom , the stiffness and inertia coefficients of which are the same 

 for both freedoms. Let us consider the general system , for which 



132) T = (Ax"+By'*), W= -(Cx* + Df), 



into which a gyrostat, or rapidly rotating symmetrical solid , is intro- 

 duced ; the direction of whose axis is determined as in the present 

 case by the coordinates x and y. (It is to be noticed that x and y 

 are principal coordinates.) The equations for the small oscillations 

 of the system are then 



By" - H z x' + Dy = 0. 



These may be treated by the general method of 45 for small 

 oscillations. In order to simplify the notation, it will be convenient 

 to put 



134) 



, A , , 



when our equations become 



8" + &,' + <*-<>, 



V'-fcl' + cfy^O. 



Having solved these, we may pass to the case of our vertical top 

 by putting c = d. 



In accordance with the method of 45, let us put 



from which we obtain 



^(* 2 + C ) 

 -4&A 



The determinantal equation is 



137) A 4 + (c 



whose roots are 



138) A 2 = {-(c + 



If the solution is to represent oscillations, all the values of A must 

 be pure imaginary, thus both values of ^ 2 must be real and negative. 



