32 Proceedings of the Royal Iruh Academy.' 



It accordingly appears that, in this simple case, although the disturbance, if 

 sufficiently small, must ultimately decrease indefinitely, yet, before doing so, 

 it may be very much increased. By taking the wave-length at right angles 

 to the direction of flow sufficiently small compared both with that in the 

 direction of flow and with the distance between the fixed boundaries, the 

 ratio of increase may be made as great as we like, provided, that is, the 

 approximate equations (29) continue to fairly represent the motion. Unless, 

 then, the limits within which these equations do hold increase indefinitely as 

 m// and mb increase, these limits may be exceeded. As Professor Love has 

 pointed out to me, the possibility of passing these limits does not afford a 

 thoroughly satisfactory proof of instability, but merely shows that the dis- 

 turbance will increase until the equations cease to represent the motion. A 

 rigorous proof that a state of motion or of equilibrium is unstable, is thus, in 

 many cases, a matter of excessive difficulty; but a result such as obtained 

 here may, I think, be regarded as strong a ];jriori evidence of instability and 

 as a satisfactory a loosteriori explanation of an actually observed instability. 



Art. 9. Practical Instcibility of Motion is co7isistent with Stability for Principal 



[Modes of Disturbance. 



At first sight, it may appear that the possibility of an arbitrary disturb- 

 ance being unstable is inconsistent with the stability of the fundamental 

 oscillations into which it can be resolved ; but, on consideration, it may be 

 seen that there is no inconsistency, and that in reality, when a system 

 possesses an infinite number of coordinates, the stability of its fundamental 

 modes of oscillation, whether about a state of steady motion or one of 

 equilibrium, affords no proof that it is stable for an arbitrary disturbance. 

 Fourier's analysis proves, in fact, that an infinite series of the type 



S (CV cos u)rt + >S',. sin w,i) 



may, at times, have values very great compared with its initial one, and may 

 even become infinite. If the question is that of the stability of a given state 

 of equilibrium, and the system possesses a potential energy-function, it is, in 

 reality, settled by the form of this function. By the well-known argument, 

 the sum of the kinetic and potential energies is constant in any motion ; and, 

 accordingly, if the latter is a minimum in the position of equilibrium, the 

 system can never deviate so far from this position that the potential energy 

 should exceed the sum of the potential and kinetic energies of the initial 

 disturbance. If we endeavour to answer the question by ascertaining the 

 nature of the roots of the equation which gives the periods of the free 



