14 DYNAMICAL THEORY OF SOUND 



to above, it is the (usually variable) moment of inertia about 

 the line of contact with the horizontal plane, provided q 

 denote the angular coordinate. 



The potential energy of the system, since it depends on the 

 configuration, will be a function of q only. If we denote it 

 by V, the conservation of energy gives 



%aq*+ F=const., ..................... (3) 



provided the system be free from extraneous forces. The 

 value of the constant is of course determined by the initial 

 circumstances. If we differentiate (3) with respect to t, the 

 resulting equation is divisible by q, and we obtain 



which may be regarded as the equation of free motion of the 

 system, with the unknown reactions between its parts elim- 

 inated. In the application to small oscillations it greatly 

 simplifies. 



In order that there may be equilibrium the equation (4) 

 must be satisfied by q = const. This requires that d V/dq = ; 

 i.e. an equilibrium configuration is characterised by the fact 

 that the potential energy is "stationary" in value for small 

 deviations from it. By adding or subtracting a constant, we 

 can choose q so as to vanish in the equilibrium configuration 

 which is under consideration, whence, expanding in powers of 

 the small quantity q, we have 



F= const. + c 2 +..., .................. (5) 



the first power of q being absent on account of the stationary 

 property. The constant c is positive if the equilibrium con- 

 figuration be stable, and V accordingly then a minimum*. It 

 may be called the " coefficient of stability." 



If we substitute from (5) in (4), and omit terms of the 

 second order in q, q, we obtain 



aq + cq = Q, ........................ (6) 



where a may now be supposed to be constant, and to have the 

 value corresponding to the equilibrium configuration. 



* In the opposite case the solution of (6) below would involve real exponen- 

 tials instead of circular functions, indicating instability. 



