430 



mod© becomes unstable if one or more of the three frequencies ceases 

 to be real. A sufficient condition for instability, under which at 

 least one frequency is pure" imaginary is/ 



A^'^^COXO. (8) 



Thus a given mode becomes unstable when 



^^'^\o)=0. (9) 



This is precisely the condition &r static buckling, which allows the 

 equilibrium equations to have nonvanishing solutions for vanishing sup- 

 plementary load. Other types of dynajnical instability of the shell are 

 of course possible with selected values of the shell parameters, but we 

 shall not examine them here. They WDuld correspond to complex frequencies 

 with nonvanishing real parts. 



When the determinant A^^^'CO), Equation (6), is expanded, and 

 terms of (^ ) and 0(^ (^) are dropped, it takes the relatively ample 

 form. 



Buckling instability in a given shell mode occurs at a critical value of 



CP , given by ^ 



t±l^. (U) 



C = C 



^ 



