184 V. OSCILLATIONS AND CYCLIC MOTIONS. 



But S is a homogeneous quadratic function of the $/s, which are 

 themselves homogeneous linear functions of the g/'s, so that S, like T a , 

 is a homogeneous quadratic function of the non- eliminated velocities. 

 Thus we have proved that the linear terms disappear from the kinetic 

 energy. At the same time we have obtained the general value of 

 the part independent of the velocities. Forming the function for 

 the kinetic potential, 



166) ^ = T' 



so that the part C which imitates the potential energy is a homo- 

 geneous quadratic function of the momenta c s of the concealed cyclic 

 motions. The terms under the sign of summation are linear in the 

 remaining velocities. 



5O. Effect of Linear Terms in Kinetic Potential. Gyro- 

 scopic Forces. We will now examine the effect of terms linear in 

 the velocities in the kinetic potential, arising from any cause what- 

 ever. We have seen that such terms arise from variable constraints, 

 and from ignored cyclic motions. We shall find a third case when 

 we treat of relative motion, 103. 



Suppose now that the kinetic potential contains the linear part 



167) <& t = L! $1 + L 2 g 2 ' H h Lm%mj 



where the coefficients L are functions of the coordinates, and may 

 also involve the time explicitly. Let the part of the force P s that 

 must be applied on account of the part d^ be denoted by P/ 1 ), so that 



168) **i)_a j l = p.M. 



dt\dq'J dq s 



Now 



and differentiating, 



We have also 



