16 Proceedings of the Roi/al Irish Academif. 



In Art. 25, p. 64, there is traced to some extent the history of a disturbance 

 whose initial type is so chosen as to make the analysis as simple as possible, 

 viz., one for which the stream-function is sin c- (7'~- - h'"'') sin sQ, h being the 

 inner radius. Only the case of one definite alternatire of the relative magni- 

 tudes is fully discussed, the choice being made so as to obtain a problem 

 sensibly different from the principal one of Chapter I. It appears that if c is 

 sufficiently large, the disturbance will increase very much before dying out. 

 The critical time is the same for all points, and an approximate expression 

 is obtained for the ratio in which the kinetic energy of the relative motion 

 throughout the whole liquid is increased at this critical time. 



One case constitutes an exception to these statements. If in the steady 

 motion the liquid rotates as a rigid body, then any small disturbance, as far as 

 terms of the first order show, neither increases nor decreases, but is simply 

 carried round with the liquid. 



It is held that as far as this investigation goes no contradiction between 

 theory and experiment is revealed. The apparent paradox that the motion 

 of a licjuid devoid of viscosity, if such existed, would be stable, while that of an 

 actual liquid of small viscosity is found by experiment to be highly unstable, 

 is disposed of by showing that though the perfect liquid may be said to be 

 stable if the disturbance is small enough, yet the limit of stability, or, to be 

 accurate, the limit within which it is legitimate to rely on equations which 

 take account of only the first powers of small quantities,^ depends on the 

 nature of the disturliance, and may be diminished indefinitely by a suitable 

 choice. And the opinions expressed by Lord Kelvin and by Eeynolds, that 

 the limit of stability of flow of a viscous liquid diminishes indefinitely with 

 the viscosity, are to some extent confirmed. Any further remarks on the efiect 

 of viscosity are postponed. 



It seems worthy of note that, as I understand it, the instability wliich is 

 actually observed in these cases may be described as a disturbance periodic in 

 time and increasing with the distance travelled by the particles rather than as 

 one periodic in distance, and increasing with the time ; that the disturbances 

 are " forced " rather than " free." I am not clear as to how far the problems 

 are analytically identical. 



^ Profes.?or Love has reminded lue of this distinction. 



