SMALL OSCILLATIONS ABOUT VERTICAL. 291 



If we call them ^ 2 , v 2 , we have for their sum and product the 

 coefficients in 137), 



139) ii* + v* = c + d + tf, [i*v* = cd. 



In order that p, v shall be real it is accordingly necessary that c, d 

 shall be of the same sign, that is our system must be either stable 

 for both freedoms , or unstable for both. Extracting the square root 

 of the second equation 139) , doubling, and adding to and subtracting 

 from the first, 



* v2 = c d b * 2 ~ /cd = ~ 



Extracting the roots, adding and subtracting, 



141) 



The inner double sign is evidently unnecessary. Since fi v = + 

 we have also, 



From the values of [i and v it is evident that both are real if c 

 and d are positive. If tehy are negative it is necessary in order to 

 have real values that 



b > ~^c + y^d. 



Thus we find that even if the system is unstable, sufficiently rapid 

 spinning of the gyrostat makes it stable. This is the case of the 

 top with its center of mass over the point of support. In order to 

 complete the discussion we have to determine the coefficients A lt A^ 

 for the various roots. If we call the roots 



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