42 DYNAMICAL THEOBY OF SOUND 



system. The coefficients a n , a 12 , a^ are in general functions of 

 q lt q 2 ; they are called the "coefficients of inertia" for the par- 

 ticular configuration considered. 



Next, let FI denote the total force acting on m, resolved 

 in the direction of s ly and let F 2 have the corresponding meaning 

 for the direction of Bs 2 . The work done on the system in any 

 infinitesimal displacement will therefore be 



2 (FM + F 2 Ss 2 ) = 2 (F&) % + 2 (F^) 8q 2 . ..... .(5) 



If there are no extraneous forces, this work is accounted for 

 by a diminution in the potential energy V of the system. When 

 extraneous forces act we have in addition the work due to these, 

 which we may suppose expressed in the form 



The coefficients Q lt Q 2 are called, by an obvious analogy, the 

 generalized "components of (extraneous) force." Hence 



. . .(6) 

 whence 



In the application to small oscillations we assume that q lt q 2 

 are small quantities vanishing in the configuration of equi- 

 librium, and for consistency we must also suppose that the 

 disturbing forces Q lt Q 2 are small. The quantities a l9 2 and 

 therefore also a n , a l2 , a& may now be treated as constants. 

 The velocity of the particle m is made up of components a^, 

 a 2 <?2 in the directions & x and 8s 2 , respectively; and if we neglect 

 the squares of small quantities its acceleration is made up in 

 like manner of components a^, ,#.,*. Hence resolving in the 

 direction of Bsi the forces acting on m we have 

 m (,& + ct 2 q 2 cos 6) = F l ,| 

 and similarly m (a^ cos 6 + o^q 2 ) = F 2 .) 



* The former of these two quantities is (to the first order) the acceleration 

 calculated on the supposition that q\ alone varies, and the latter is the accelera- 

 tion when #2 alone varies. It is only on the hypothesis of infinitely small 

 motions that the resultant acceleration is obtained by superposition of these. 



