165] SMALL OSCILLATIONS. 267 



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

 quadratic function of the generalized velocities q 1} g^,..., say 



l q 2 + ............ (1). 



The coefficients in this expression are in general functions of the 

 coordinates q l} 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, <? 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 l} q 2 , ... , with (on the same understanding) constant 

 coefficients, say 



2 + ............ (2). 



By a real* linear transformation of the coordinates q lt q 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 Oj, a 2 , ... are called the 'principal coefficients of 

 inertia'; they are necessarily positive. The coefficients c 1? 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 

 Aji, Ag 2) ... may be expressed in the form 



Q 1 Ag 1 + Q a Ag a + ........................ (5). 



The coefficients Q lt Q 2 , ... 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 (1^)), 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. 



