﻿subjected to Forces in the Direction of their Axes, 239 



The last terms in the numerator and denominator o£ this 

 expression are very small and not sufficiently accurate to be 

 retained except as a step in the third approximation i£ such 

 were required. 



Hence the frequency should be given with considerable 

 accuracy by 



-^=500-4 -?£+ 12-153 X 



Hi pcor pool 



500-4— n+ 12-153- h . . . . (11) 



™or owl- v ' 



The terminal couples are given in this approximation by 



M=-079158(BI+^ c V-71145p«»Py l +8-598dPy 1 ) 



1GP/ 2 

 gj-10-*C66137^«Pyi + 7"9365Py!). 



§ 10. Another approximation may be obtained by the 

 dynamical method, in which a type of vibration is assumed *« 



Thus, taking equation (10) for the type, the potential 

 energy at any instant is 



v-aif(g)'- +i r£(^ 



/ 512EI 256 P \ a 

 "1 5P + 105 l) yii 



and the kinetic energy of the system is 



The equation 



dt + dt 



^ iYeS _^ = 501 E ^+12-^ (12) 



y x poor poor ' 



A solution which is based on a very reliable method, but 

 necessarily overestimates the value of — •— , since equation (10) 

 does not exactly represent the true type of displacement. 

 * See Eayleigli's ' Theory of Sound.' vol. i. § 182. 



