158 V. OSCILLATIONS AND CYCLIC MOTIONS. 



If the equilibrium is stable the potential energy must be a minimum 

 so that the constants c rs will be such that the quadratic function W 

 is positive for all possible values of the variables q. 



The kinetic energy will be a quadratic function of the time 

 derivatives, q[, q' 2 , . . . qi, 



58) T= 



where the a's are functions of the coordinates q alone. We may 

 develop the functions a rs in series, thus one term of the sum becomes 



59) ctr.qlq,' = 



*=1 



and since the velocities q' are small at the same time as the co- 

 ordinates g, we may neglect all the terms within the braces except 

 that of lowest order aj,, therefore we may consider the a's as 

 constants. If we have besides the conservative forces of restitution, 

 arising from the potential energy W, non- conservative resistances 

 which are linear functions of the velocities, we may make use of a 

 dissipation function F, 39, such that the dissipative force correspond- 

 ing to the coordinate q r will be - - ~ 7- We thus have the three 

 homogeneous quadratic functions with constant coefficients, 



rn = ! 



60) * = -g 



r=l * = : 



Each of these has the property of being positive for all possible 

 values of the variables of which it is a function. The a's may be 

 called coefficients of inertia, the c's, coefficients of stiffness, and 

 the jc's, coefficients of viscosity or resistance. We may now form 

 Lagrange's equations for any coordinate q r . 



dp r dT dW dF 



^t~W r ~^ = ~W r ~Wr 



where 



