70 Proceedings of the Royal Irish Academy. 



Chapter I., pp. 80-94, deals with Lord Kelvin's investigations * 



The two problems which he discussed having been described in Art. 1, 

 p. 80, an abstract is given in Art. 2, pp. 80-83, of one of his proofs 

 that an infinitely wide stream of finite depth and uniform vorticity is stable ; 

 this solution, following Lord Eayleigh, I describe as a " special " solution 

 in contradistinction to another which he indicated in a subsequent paper. 

 As far at least as the velocity-component in the direction of the depth 

 is concerned. Lord Kelvin first obtains a solution, (v), of the differential 

 equation which satisfies the most general initial conditions throughout, but 

 violates the permanent boundary-conditions at the top and bottom of the 

 stream; he then adds to this solution a " forced " disturbance, (to), which 

 would be caused throughout the stream by exactly reversing this outstanding 

 boundary disturbance, and, by addition, thus obtains a solution which does 

 satisfy the boundary-conditions. The " forced " disturbance is obtainable 

 as an integration of an infinity of constituents each of which is simply- 

 periodic in the time, and the constituents are to be chosen by a Fourier 

 analysis, valid between the times t = - co and t = + <x> so as to satisfy 

 the boundary-conditions to = from t = - co till t = Q, and to = - v from 

 ^ = till t = CO . The V solution is composed of one or more terms, each 

 of which has a factor which involves the time exponentially, the index 

 being essentially negative, and eventually varying as the cube of the time ; 

 thus V diminishes indefinitely; and Lord Kelvin states that hence the "forced" 

 disturbance to, which rises gradually from zero at ;^ = 0, also diminishes 

 indefinitely, and concludes that the steady motion is stable. 



Art. 3, p. 83, contains a brief account of another proof of stability in 

 the same motion, which Lord Kelvin indicates in his discussion of the second 

 of the two problems which he discussed. 



Art. 4, p. 84, gives Lord Eayleigh's adverse criticism of the second solution, 

 in which he points out that Lord Kelvin has merely shown the possibility 

 of obtaining forced vibrations of arbitrary (real) frequency, and that this 

 constitutes no proof of stability, it being possible to do this in the case of 

 a pendulum displaced from a position of unstable equilibrium. 



Art. 5, pp. 84-85, gives remarks by Lord Eayleigh on the " special " 

 solution in which he appears to accept it. 



In Art. 6, p. 85, it is pointed out, however, that the " special " solution 

 involves a tacit assumption that the " forced " disturbance, to, vanishes 

 everywhere throughout the liquid at the time t = 0. 



In Art. 7, p. 86, it is argued that this assumption is legitimate if it 



•*Phil. Mag., August and September, 1887. 



