Oer — Stability or Instability of Motions of a Perfect Liquid. 11 



Eeynolcls also has attacked theoretically the latter of the two problems 

 discussed by Lord Kelvin, and has obtained an inferior limit to the ^'elocity 

 for which the motion can be unstable^; his result is of the same order of 

 magnitude as that which he obtained experimentally in the somewhat 

 different case of flow through a pipe. 



An inferior limit, different from that of Eeynolds, but of the same order 

 of magnitude, has been obtained theoretically by Sharpe," who has also 

 deduced, in the case of flow through a pipe, a limit of the order of that 

 observed by Eeynolds. 



In the case of the liquid shearing uniformly, H. A. Lorentz=* has obtained 

 a limit which is of the same order. 



Both these writers use Eeynolds' method. 



The investigation here presented deals exclusively with questions in 

 w^hich viscosity is altogether ignored. 



Its contents may be summarized as follows : — 



In Chapter I., pp. 17-42, cases of motion in plane strata are discussed. 



In Art. 1, p. 17, a brief outline is given of Lord Eayleigh's investigation of 

 the fundamental free disturbances, reference being made to Lord Kelvin's 

 objection, which I confess I do not understand, and to Lord Eayleigh's 

 reply thereto. 



In Art. 2, p. 20, 1 have given what appear to be the most important portions 

 of Love's criticism of these investigations, and have remarked upon them in 

 Art. 3, p. 22. In brief. Professor Love has made three objections to Lord 

 Eayleigh's solution, viz.: (1) the free disturbances involve slipping in the 

 interior of the fluid ; (2) the wave-velocity is restricted wdthin certain limits ; 

 (3) it has not been shown that an arbitrary disturbance can be replaced by 

 a system of Lord Eayleigh's type. Of these it appears to me that (1) and 

 (2) have no force whatever, but that (3) -calls for further examination. 



In Art. 3a, p. 23, I point out, however, that in Lord Eayleigh's free 

 disturbances, although the velocity at a given point, as given by terms of the 

 first order of small quantities, is periodic in time, yet the amplitude of the 

 waves generally increases ; owing to this, in itself, his conclusion as to 

 stability may require modification ; but this cannot be decided without 

 taking into account terms of smaller order. 



In Art. 4, p. 24, Professor Love's third objection is considered ; and taking 



1 " On the Dynamical Theory of Incompressible Viscous Fluids, and tlie Deterniinaiiou of th. 

 Criterion," Phil. Trans., A, t. clxxxvi., Part i., p. 123 (1895) ; Sc. Papers, t. ii., p. .535. 



2 Trans. Anier. Math. Soc, Oct., 1905. 



* Abh..ndlungen uDer theoretisclie Physik, Bund i., b. 'il'6. 



