27-29] Rotating Systems 31 



than its value when at rest in the equilibrium configuration by a small 

 constant amount c. Thus throughout the subsequent motion T R can never 

 increase beyond the value c, so that the motion is absolutely stable. This 

 argument cannot however be reversed to shew that the system is necessarily 

 unstable if W ^o> 2 ./ is not an absolute minimum. 



Let us examine what happens when the relative motion of the system is 

 affected by dissipative forces, such as viscosity. The right hand of equation 

 (40) will be negative except when the system is relatively at rest, so that 

 T R + W ^o)' 2 I will decrease indefinitely. If W - \<?I was an absolute mini- 

 mum in the position of equilibrium, this condition can only be satisfied by 

 T R being reduced to zero, and the system coming to rest in its position 

 of equilibrium. But if W ^o> 2 / was not an absolute minimum in the con- 

 figuration of equilibrium, there will be a possible motion in which W ^o> 2 / 

 continually decreases while T R remains small at first, but may increase 

 beyond limit when W |&> 2 / is sufficiently decreased. The system is now in 

 a restricted sense unstable. 



Instability of the kind just discussed is called "secular instability." The 

 conception of "secular instability" was first introduced by Thomson and 

 Tait*. It has reference only to rotating systems or systems in a state of 

 steady motion ; for systems at rest secular stability become identical with 

 ordinary stability. It is clear that a system which is ordinarily stable may 

 or may not be secularly stable, but a system which is ordinarily unstable 

 is necessarily secularly unstable. 



Mass rotating freely in space 



29. As Schwarzschildf has shewn, the conditions of secular stability 

 assume a somewhat different form for a mass rotating freely in space. Here 

 the rate of rotation is not constant but varies with the moment of inertia of 

 the mass ; if we refer the motion to axes rotating with a uniform velocity 

 the rotation of the freely rotating mass may lag behind that of the axes and 

 the relative coordinates #, y, z may increase without limit although the 

 configuration remains stable. It is therefore important to express the con- 

 ditions of stability in a form which does not involve the constancy of o>. 



When the mass is rotating freely in space, G = so that (equation (30)) 

 M is constant. The elimination of o> from equations (25) and (26) leads to 



T-T + 

 ^ + 2/ 



where T s =T R -j?. 



* Nat. Phil. 2nd Ed. n. p. 391. * 



t See Schwarzschild; "Die Poincare'sche Theorie des Gleichgewichts." Neue Annalen. d. 

 Stermcarte Miinchen, 3 (1897), p. 275, or Inaugural Dissertation. Miinchen (1896). 



