12 Proceedings of the Royal Irish Academy. 



the simplest possible case, that of a liquid which is shearing uniformly, it is 

 shown that, at least in this case, the most general disturbance can be resolved 

 into a series of the type obtained by Lord Rayleigh. The resolution is 

 effected for an example of the simplest type analytically, i.e., one in which the 

 initial velocity-components are sine-cosine functions of the coordinates ; and 

 when the initial disturbance is of this character an expression is obtained for 

 the velocity at right angles to the bounding-planes at any time. 



In Art. 5, p. 26, this same result is obtained more directly from the funda- 

 mental equations without reference to Lord Eayleigh's " free modes." When 

 the disturbance is three-dimensioned, the expressions for the velocities parallel 

 to the bounding-planes involve transcendental integrals, and accordingly 

 the complete solution is given for the two-dimensioned case only. 



The solution thus obtained is periodic in the direction of flow and of 

 assigned wave-length ; in Arts. 6, 7, pp. 28, 29, it is indicated how the 

 solution is to be modified in two other instances in which other and more 

 definite conditions are to be satisfied at the C7ids of the stream. 



In Art. 8, p. 29, the solution which has been obtained is examined ; and it 

 is readily seen that if the initial wave-length perpendicular to the bounding- 

 planes is small compared with the wave-lengths in the directions parallel to 

 them, and also small compared with the distance between them, the original 

 disturbance increases and attains a maximum value, much greater than its 

 initial, at a certain critical time, after which it diminishes without limit. For 

 the two-dimensioned case, the order of the increase can be stated in a simple 

 form in two extreme cases : — if the wave-length in the direction of flow is large 

 compared with the thickness of the stream, the ratio in which the kinetic 

 energy of the relative motion increases is of the order of the square of the 

 number of wave-lengths perpendicular to the stream which are contained in 

 the original disturbance ; while if the wave-length in the direction of flow is 

 small compared with the thickness of the stream, the ratio of increase is of the 

 order of the square of the ratio of the wave-length in the direction of flow to 

 that perpendicular to the boundaries. This constitutes, I think, a satisfactory 

 explanation of the instability which observations of motion in pipes lead us 

 to expect also in cases of plane stratified flow. 



In Art. 9, p. 32, it is pointed out that coexistence of the stability or 

 neutrality, established by Lord Eayleigh, in the case of each of the fundamental 

 modes of disturbance, with what may, I think, be described as practical insta- 

 bility for others of a more general type is quite in keeping with the teaching of 

 Fourier analysis ; that the question of the stability of a state of equilibrium is 

 in reality decided by a potential-energy criterion ; and that the light thrown 

 on the question by a knowledge of the reality of the " free periods '' is only 



