28 General Dynamical Principles [on. n 



so that ffgr = - &. and . = 0. Then 



Using the value of T 7 given by equation (24) and keeping o> constant, 

 we have 



^/ar\_8T_^/aTfi\_aT fl^-^l-i ^ 

 dt ( d ej w. " * V 80; / " aft 4 * I <** W 8 ^J- ' a ^ 



so that the equation of motion (29) becomes, using (31), 



Thus the equations of motion relative to rotating axes differ from the 

 simpler equations appropriate to the case of CD = in two respects ; first by 

 the presence of what we may call "gyroscopic" terms such as @ ls (o0i, and 

 second, that W - ^&> 2 / replaces the potential energy W of the simpler 

 equations. 



26. The conditions for equilibrium relative to the moving axes are 



^ = ^=...=0 

 and so are determined by the equations 



g|(ir-irt) **.<* ......................... (33), 



reducing when there are no externally applied forces, to 



^(F-K/) = .......................... (34). 



The difference between these equations and the simpler ones for a system 

 at rest is merely that W has become replaced by W |o) 2 /. The configurations 

 of relative equilibrium may accordingly be found just as though the system 

 were at rest under a potential W J&> 2 /, and these configurations will fall 

 into linear .series as before. 



27. To discuss the small oscillations of such a system, let us return to 

 the equations of motion (30), and suppose we are considering the oscillations 

 of a configuration which is one of equilibrium under no applied forces, say 



0, = @j, etc. 



Let the coordinates be replaced by 6 1 lt etc. so that the new values 

 of O lt &z, ... all vanish in the configuration of equilibrium. The values of 



