197] CONDITION OF SECULAR STABILITY. 327 



shewing that under the present supposition neither ^ nor V T Q 

 can increase beyond a certain limit depending on the initial 

 circumstances. 



Hence stability is assured if F T is a minimum in the 

 configuration of relative equilibrium. But this condition is not 

 essential, and there may even be stability with V T a maximum, 

 as will be shewn presently in the particular case of two degrees of 

 freedom. It is to be remarked, however, that if the system be 

 subject to dissipative forces, however slight, affecting the relative 

 coordinates q lt q z , ..., the equilibrium will be permanently or 

 'secularly' stable only if F- T is a minimum. It is the 

 characteristic of such forces that the work done by them on 

 the system is always negative. Hence, by (7), the expression 

 & + ( T ) will, so long as there is any relative motion of the 

 system, continually diminish, in the algebraical sense. Hence if 

 the system be started from relative rest in a configuration such that 

 V T is negative, the above expression, and therefore d fortiori 

 the part V T , will assume continually increasing negative 

 values, which can only take place by the system deviating more 

 and more from its equilibrium-configuration. 



This important distinction between ' ordinary ' or kinetic, and 

 ' secular ' or practical stability was first pointed out by Thomson 

 and Tait*. It is to be observed that the above investigation pre- 

 supposes a constant angular velocity (ri) maintained, if necessary, by 

 a proper application of force to the rotating solid. When the solid 

 is free, the condition of secular stability takes a somewhat different 

 form, to be referred to later (Chap. XII.). 



To examine the character of a free oscillation, in the case 

 of stability, we remark that if \ be any root of (10), the equations 

 (9) give ' 



-^8 _ ^3_ _ fl /-, 0\ 



- 



where A rl , A r2 , A r3 , ... are the minors of any row in the determi- 

 nant A, and (7 is arbitrary. It is to be noticed that these minors 

 will as a rule involve odd as well as even powers of \, and so 



* Natural Philosophy (2nd ed.), Part i. p. 391. See also Poincare, "Sur 

 I'e'quilibre d'une masse fluide animee d'un mouvement de rotation," Acta Mathe- 

 matica, t. vii. (1885). 



