36 Proceedings of the Royal Irish Academy, 



It appears then that the body may be such, and so moving, that, in spite 

 of the reality of the free periods, a small initial disturbance of the steady 

 motion may lead at some time to a large one, that is as far as can be 

 ascertained by equations which take account only of the first powers of small 

 quantities. 



In this example, as well as in the preceding one relative to a state of 

 equilibrium, the value of k cannot, of course, ever he equal to unity exactly; 

 in this limiting case the values of iJ, q become equal, and the solution of the 

 equations of motion assumes a different form;* so that another mode of 

 contrasting the case of equilibrium when there is an energy-function with 

 those of equilibrium when there is no energy-function and of steady motion 

 is to say that, in the former case, equality of periods cannot be destructive of 

 stability, but in the others it may ; and also that in the others the evil effects 

 of what may be regarded as in reality an approach to equality of periods 

 cannot be estimated by regard to the ratio of the periods alone. And in this 

 connexion it may be borne in mind that, in the liquid system under discus- 

 sion, we have an extreme case of the equality of free periods, as their values 

 range continuously from one limit to another. 



Akt, 10, The hearing of the Non- Vanishing of the Integrated Product of 

 Velocities in tivo Principal Modes. 



The fact that for some types of disturbance the steady motion may 

 be practically unstable in spite of the stability of the fundamental modes 

 may be seen to be connected with another fact noted above (p. 24), namely, 

 that if Vi cos \{x - Uit), Vz cos \{x - TJf) denote the values of v in two 

 fundamental modes having the same wave-length in the ^-direction, we do 

 not have, as in the case of a system possessing a potential-energy function 

 and oscillating about a position of equilibrium, the relation 



6 



ViV^dy = 0. 







If this relation did hold, it is easily seen that the total kinetic energy due 

 to the velocity in the ^/-direction would be independent of the time, whereas 

 in the actual case there occur terms of the type 



cos 

 sin 



\{U,- U,)t 



ViVzcly, 



whose value at the time t may be large compared with their initial value. 



* And in this limiting form the expressions for the coordinates contain terms proportional to 

 t cos pt, t sinjui ; equalitj^ of periods introduces no such terms into the solution of the equations-of 

 the small motions of a system displaced from equilibrium if it possesses an energy- function. 



