354 Mr Basset, On the Steady Motion [May 30, 



Let any disturbance be communicated to the system, and let 

 E + BE be the energy of the disturbed motion ; then 



E+8E = X + $+V, 

 whence 



8E=Z + [(® + V)-(® +V )] (9). 



Now X, being the kinetic energy of a possible motion of the 

 system, is essentially positive, and in the beginning of the dis- 

 turbed motion must be a small quantity ; hence if 



$+V>® +V , 



% must remain a small quantity, and the system cannot deviate 

 much from its position in steady motion, and the motion will 

 be stable ; but if 



®+v<$ +v , 



% may become a finite positive quantity, whilst the term in square 

 brackets may become a finite negative quantity, such that their 

 difference remains equal to the small quantity BE, and the motion 

 may be unstable. Hence the motion will be stable, provided 

 i? + V is a minimum. 



It should be noticed, that this criterion of stability not only 

 includes disturbances which produce variations of the coordinates 

 6, but also disturbances which produce variations of the momenta 

 k, though of course the latter quantities always remain constant 

 during the disturbed motion, and equal to their values immediately 

 after disturbance. 



6. Routh has shown*, that when there is only one coordinate 

 of the type 0, the steady motion will be unstable unless £ + V is a 

 minimum in steady motion ; but although we have just shown, 

 that when there are two or more coordinates of the type 6, the 

 motion will be stable provided $ + V is a minimum, it does not 

 follow that the motion will be unstable, when this condition is not 

 satisfied. In fact it sometimes happens, that steady motion will 

 be stable when £ + V is a maximum. To see this, let us re- 

 turn to the case of the ellipsoid rotating about its centre. 



When the axes of rotation are the least, greatest and mean 

 axes, the values of £ in the beginning of the disturbed motion 

 are respectively equal to 



2 /c 



C - {(G - A)cos 2 cf> + (G - B)sin 2 cf>}6 2 ' 

 A + (B-A)<f> 2 + (C-A)e 2 ' 



2 K 



B-(B-A)f 2 + (G-B)e 2 ' 



* Stability of Motion, p. 85. 



