20 Proceedings of the Royal Irish Academy. 



In a later paper, which refers chiefly to motion through a circular pipe, 

 Lord Eayleigh points out that, if n be complex, there is no " disturbing 

 infinity," and that therefore his argument does not fail if regarded as one 

 for excluding complex values of n, though what happens when n has a value 

 such that n + kU vanishes at an internal point, is a subject for further 

 consideration* 



To this subject he returns ; and both in the case in which the vorticity 

 in the steady motion is constant through certain layers, but discontinuous at 

 their boundaries, and that in which it is continuous throughout but varying, 

 he concludes that the infinities which present themselves when n + kU is zero, 

 do not seriously interfere with the application of the general theory, so long 

 as the square of the disturbance from steady motion is neglected.f 



And, in his latest paper on the subject, taking the simple case in which 

 in the steady motion the velocity increases uniformly from each wall to the 

 centre of the stream, he has examined the effect of including in the investi- 

 gation the squares and higher powers of the small quantities as far as the 

 fifth power. He concludes that there is no sign of the amplitude of a wave 

 tending spontaneously to increase, as far as his investigation goes.J His 

 discussion is, however, limited to the very restricted class of disturbances 

 which do not involve any slipping at the surface where the vorticity is dis- 

 continuous. And if such slipping be introduced, the contrary result would 

 apparently be arrived at. (See Art. 3b, below.) 



Art. 2. Prof. Love's Criticism of the above. 



In a criticism of these investigations of Lord Eayleigh, Professor Love 

 writes§ [having replaced n/k by - V, so that equation (7) becomes 



{U- V) [d^v/df - k'v) - vd' U/df = (14)] :— 



" In order that the disturbance may be propagated by waves in the 

 manner supposed, it must be possible to assign a real quantity V so that a 

 function v may exist which (i) satisfies the differential equation [14, above] 

 for all values of 3/ in a certain real interval, (ii) vanishes at the limits of this 

 interval, (iii) is finite, and has a finite and continuous differential coefficient 



* " On the Question of the Stability of the Flow of Liquids," Phil. Mag., t. xxxiv., p. 59, 1892 ; 

 Scientific Papers, t. iii., p. 581. 



t "On the Stability or Instability of Certain Fluid Motions," iii., P. L. M. S., xxvii., p. 11, 

 1895; Collected Papers, iv., pp. 207, 208. 



X "On the Propagation of "Waves upon the Plane Surfaces separating two Portions of Fluid of 

 Different Vorticities," P. L. M. S., xxvli., 1895 ; Collected Papers, iv.— the concluding sentence. 



§ " Examples illustrating Lord Kayleigb's Theory of the Stability or Instability of certain 

 Fluid Motions," Proc. L. M. 8., Jan. 9, 1896, xxvii., p. 202. 



