165] SMALL OSCILLATIONS. 267 



energy T will, as explained in Art. 133, be a homogeneous 

 quadratic function of the generalized velocities (ft, g^,..., say 



2r=a n 2 1 2 + a 22 g 2 2 +... + 2a 12 ? 1 ? 2 + ............ (1). 



The coefficients in this expression are in general functions of the 

 coordinates &amp;lt;ft, q 2 ,..., but in the application to small motions, we 

 may suppose them to be constant, and to have the values corre 

 sponding to (ft = 0, q 2 = 0,.... Again, if (as we shall suppose) the 

 system is conservative, the potential energy V of a small displace 

 ment is a homogeneous quadratic function of the component 

 displacements q 1} (? 2 , ... , with (on the same understanding) constant 

 coefficients, say 



... + 2c 12 g 1 g 2 + ............ (2). 



By a real* linear transformation of the coordinates (ft, ^ 2 ,... it 

 is possible to reduce T and V simultaneously to sums of squares ; 

 the new variables thus introduced are called the normal 

 coordinates of the system. In terms of these we have 



(3), 

 (4). 



The coefficients a 1} a 2 ,... are called the principal coefficients of 

 inertia ; they are necessarily positive. The coefficients c l5 c 2 ,... 

 may be called the principal coefficients of stability ; they are all 

 positive when the undisturbed configuration is stable. 



When given extraneous forces act on the system, the work 

 done by these during an arbitrary infinitesimal displacement 

 A(ft, Ag 2 , ... may be expressed in the form 



(5). 



The coefficients Q lt Q 2 &amp;gt; are then called the normal components 

 of external force. 



In terms of the normal coordinates, the equations of motion 

 are given by Lagrange s equations (Art. 133 (17)), thus 



_____ 



dtdq s dq s ~ dq s s 



* The algebraic proof of this involves the assumption that one at least of the 

 functions T, V is essentially positive. 



