0\i\i—StahiUty or Instahility of Motions of a 'Perfect Liquid. 15 



desirable to give of the approximations used renders this and some succeeding 

 portions of the discussion somewhat tedious. 



In Art. 19, p. 53, it is shown that this initial disturbance, and any other 

 in which the velocities have a definite wave-length in the direction of flow, 

 must eventually diminish indefinitely according to the same laws as in the 

 plane stratified case. 



In Art. 20, p. 54, another instance of initial disturbance is considered in 

 which the radial velocity is sin m (r- - If) sin Tcz. The wave-length along the 

 pipe is supposed small compared with the outer radius, a result similar to that 

 deduced for the former example being obtained. In this case, however, 

 the critical time is the same at all points ; and accordingly an approximate 

 expression is obtained for the ratio of increase of the energy of the relative 

 motion throughout the whole pipe at this critical time. 



In Art. 21, p. 58, the initial disturbance of the preceding Article is discussed 

 under a different extreme supposition, viz., that the wave-length along the 

 pipe is large compared with the outer radius ; and similar conclusions 

 are drawn. 



Although quantitative comparison is easier in the extreme cases of waves 

 which are long and of waves which are short in the direction of flow, there 

 is reason to suppose that a disturbance of any wave-length whatever in this 

 direction, if of much shorter, and sufficiently short, wave-length in the 

 direction at right angles, will increase very much, provided equations remain 

 valid in which the squares of small quantities are neglected. 



Chapter III., pp. 61-68, discusses steady motion in cylindrical strata, 

 rotating round a common axis. 



Art. 22, p. 61, deals with Lord Eayleigh's brief reference to this case. 



The analysis appropriate to the investigation of the two -dimensioned 

 disturbances which are harmonic functions of the time is given in Art. 23, 

 p. 61. It is seen that the only law of flow for which the solution can readily 

 be obtained is that obeyed by a viscous liquid when one or both of the 

 cylindrical boundaries are made to rotate. The solution again involves 

 slipping in the interior as well as at the boundaries. It is shown how the 

 most general two-dimensioned disturbance can be propagated by means of 

 elementary ones of the type obtained, and how the result to which this 

 resolution leads may be obtained directly, without reference to the funda- 

 mental free modes. 



In Art. 24, p. 63, it is shown that any two-dimensioned disturbance, in which 

 initially the relative velocity components vary as coss0, sins0, .9 being a 

 definite number, will eventually diminish indefinitely according to laws 

 similar to those which hold for the cases discussed in the preceding chapters. 



