26 General Dynamical Principles [OH. n 



Combining this with the result obtained in the last section, it appears 

 that a change of stability occurs at every point of bifurcation, and at every 

 point on a linear series at which /JL passes through a maximum or a minimum 

 value, agreeing with the result obtained by other means in 19 and 20. 

 The criterion of stability in the branch series at a point of bifurcation is 

 most readily seen by the method already adopted in 21 ; with the con- 

 ventions there used, it appears that the branch series will be stable if it 

 turns upwards from the point of bifurcation, and unstable if it turns down- 

 wards. 



ROTATING SYSTEMS 



24. This completes the discussion of the stability of statical systems. 

 The stability of motion of a dynamical system is a very much more com- 

 plicated question, but assumes a specially simple form when the motion 

 consists mainly of a rigid body rotation. We proceed to discuss the stability 

 of such a system. 



Let the system be referred to axes rotating in space with any velocity co 

 about the axes of z in the direction from Ox to Oy. Let x, y, z be the 

 coordinates of any point referred to these axes, and let x, y, z denote their 

 rates of increase. The components of velocity in space are then given by 



u x yw, v = y + xco, w = z ..................... (19) 



so that the kinetic energy T is given by 



= J2ra (& + y 2 + z 2 ) + o>2m (xy - yx) + J<o 2 2m (a? + f) ...... (20). 



The total moment of momentum M about the -axis is given by 

 M = Sm (xv yu) 



= 2m (xy - yx) + o>2w (a 8 + f) .................. (21). 



Put 



T R = &m(&-+F + #) ........................ (22), 



U = ^m(xy-yx) .............................. (23), 



/ = 2m(# 2 +t/ 2 ) .............................. (24), 



so that T R is the kinetic energy relative to the rotating axes, U is the 

 moment of momentum relative to the moving axes, and / is the moment of 

 inertia. Then equations (20) and (21) become 



T = ^+o>/ + ia> 2 / ........... '. ............ (25), 



M= U+ col ........................... (26). 



Eliminating U we obtain 



GfiI ........................... (27). 



