SMALL OSCILLATIONS 351 



Lagrange's equations, in terms of these coordinates, are 



d /dT\ dT dW 



- 7TT- = TTT- ' etc -> 



which become ft = a^ lt etc. (197) 



Stable Equilibrium 



282. If ^ is positive, let us put - = &J, so that ^ will be 

 real. The equation is now 



of which the solution is 



^ 1 = -4 1 cos(^ j, (198) 



as in 208. Thus the motion is a simple harmonic motion of 

 frequency k r If all the coefficients a lt a v , a n are positive, the 

 complete solution of the equations will be of the form 



2 ), etc., 



and the coordinate x of any particle, of which the value in the 

 equilibrium position is a? , will be 



dx 



where ^, 5 2 , are new constants. 



Thus the motion of any single particle will be a motion com- 

 pounded of a number of simple harmonic motions. 



