Okr — Stability or Instability of Motions of a Perfect Liquid. 13 



indirect. The case of a system possessing only two coordinates is considered ; 

 and it is shown that if there is no potential-energy function, stability, or 

 rather neutrality, of the two fundamental modes is quite consistent with very 

 narrow limits of stability for a combination of both. Wlien the question is one 

 of the stability of a state of motion, it does not appear to have been established, 

 for a system possessing an infinite number of coordinates, that reality of 

 the periods of the fundamental disturbances necessitates stability for an 

 arbitrary disturbance, however small, even when there is a potential-energy 

 function. A concrete instance — that of an unsymmetrical spinning-top 

 standing up and acted on by gravity — is given, in which the reality of the two 

 fundamentable periods is compatible with practical instability for a more 

 general disturbance. It seems only another mode of contrasting these cases 

 to assert that equality of two periods cannot affect the stability of equilibrium 

 of a system possessing an energy-function ; but that equality of periods may 

 destroy, and approximate equality may endanger, the stability of a state of 

 motion, and that, moreover, the extent of the danger cannot be judged by a 

 mere comparison of the periods. 



In Art. 10, p. 36, it is pointed out how the impossibility of inferring stability 

 in general from that of the fundamental disturbances is connected with the fact 

 that the latter do not possess the property characteristic of the oscillations 

 about a state of equilibrium of a system having a potential-energy function, 

 viz. .-—that the integrated product of the corresponding velocities in any 

 two principal modes vanishes. 



In Art. 11, p. 37, it is shown from the solution obtained that if the end- 

 conditions are such that the velocity components are periodic in the direction 

 of flow, the energy of the actual as well as of the relative motion increases 

 for a time, and that this arises from work being done by the pressures, which 

 cannot be strictly periodic in the direction of flow. 



In Art. 12, p. 38, it is shown that any disturbance of an ordinary type 

 must remain finite, and that in the most general one, provided the velocities 

 possess a definite wave-length in the direction of flow, the relative velocity 

 component in that direction, as determined by the solution given, eventually 

 diminishes indefinitely, varying inversely as the time, while the component at 

 right angles eventually varies inversely as the square of the time, so that it 

 may be said the steady motion is stable, provided the initial disturbance is 

 small enough. 



Art. 13, p. 39, deals briefly with the more general case of a stream composed 

 of a number of plane layers, each of which is shearing uniformly, but at a 

 rate which is different in different layers. The solution of even the two- 

 dimensioned prol)leni cannot readily be gi\'en in a form which admits of 



