THEORY OF VIBRATIONS 43 



If we multiply the former of these equations by ^ and the 

 second by a^, and sum for all the particles of the system, we 

 find, with the notation of (4), 



dV . 





.(9) 



and similarly a^ + a&q 2 = 5 h Q 



oq% 



where 021 is of course identical with a 12 . 



When there are no extraneous forces these equations are by 

 hypothesis satisfied by q^ = 0, q a = 0. The configuration of 

 equilibrium is therefore characterized by the property that 



i- =0 ' f-- (10) 



in other words, the potential energy is stationary for all infini- 

 tesimal displacements therefrom. Hence if V be expanded in 

 powers of q lt q 2 , the terms of the first order will be absent, and 

 we may write with sufficient approximation 



2 V = C-aq? + 2c 12 <? 1 <7 2 + Ca^jj 2 , (11) 



a constant term being omitted. The quantities c u , c 12 , c^ are 

 called the " coefficients of stability." 

 Hence (9) may be written 



a^i + ^22^2 

 where c 21 = c 12 . 



If we look back to any of the special problems of 14 we 

 shall recognize that the equations of motion are in fact of this 

 type. For example, in the case of the double-pendulum we 

 have 



...(13) 



The formulae therefore correspond if we put 



ft = ar, q* = y, \ 



a n = M, 12 = 0, 022 = m, V ...(14) 



c n = (M + m) g/a + mg/b, c u = - mg/b, C& = mg/b.J 



