﻿in the Double Pendulum. 383 



other bob in one mode is the reciprocal of the corresponding 

 ratio in the other mode/' 



Proceeding to the general case of an elastic system with 

 two degrees of freedom, using Professor Lamb's notation, 



2T = A6 2 + 2U0cf> + B4>\ 



2 V = 2V + ad' 2 + 2h6<f> + b(f> 2 j 



so that with a time factor = in 



7i-(A0+H<£) = a6 + l«f>, 



?r(H<9 + B(£) = 7,0 + ty; 



whence the product of the roots in 6j^> is Ti — / & • 



If H = so that T is a function of squares of velocities, 

 the product of the amplitude ratios is — B/A, or, in the 

 double pendulum, — M/m. 



If A = so that the potential energy is a function of 

 squares of displacements, the product of the amplitude 

 ratios is bja, i. e. the ratio of the two stability coefficients. 

 Thus in either case we have an energy relation. For the 

 kinetic energy take 



H = and ££=-?; 



9192 A 



•square and multiply by A 2 /B 2 , and insert the frequencies. 

 For the potential energy take 



A = and -V = — - 



0102 



i 

 2 <* 



which, when squared as before, yields a similar relation. 

 The two results may be expressed in words thus : — 



When the Kinetic or Potential Energy is written as a 

 ■function of squares only, the ratio of the Kinetic or 

 Potential Energy expressed in one co-ordinate to that 

 •expressed in the other co-ordinate for one normal mode is 

 the reciprocal of the corresponding ratio for the other 

 normal mode. 



This investigation gives an insight in certain cases into the 

 indeterminateness of the normal modes with equal periods. 



