Oer — Stahility or Instahility of Motions of a Perfect Liquid. 37 



Similarly, in the two-dimensioned problem, the kinetic energy due to 

 ^-component of relative velocity would be independent of the time, provided 



ih{n,idy = 0, . 



or 



ody'dy' 



which relation does not actually hold. 



And, in the two-dimensioned problem, the total kinetic energy of the 

 relative motion involving both x and y components of velocity would be 

 independent of the time, provided for every two fundamental modes, 



Jo ]o\d^ dx dy dy J 



This integral is seen to reduce to 



d-ipi 









dy 



where | dip^/dy \ denotes the discontinuity in the value of d-ipz/dy at the 

 plane where slipping occurs in the corresponding fundamental disturbance, 

 and accordingly does not vanish as a rule. 



Akt. 11. Energy of Actual Motion increases ; Work is done hy End-Pressures. 



It has been shown that the kinetic energy of the relative motion of a 

 disturbance may increase, and the same is true for the energy of the actual 

 motion. When the disturbance is periodic in x, the energy of the total 

 motion is, in fact, equal to that of the steady motion, together with that 

 of the relative motion. The difference in fact is W^yudxdy; and in 

 estimating this correctly to the second order of small quantities, terms of the 

 second order in u must be taken account of. To whatever order, however, 

 the approximation is made, this integral is zero if taken through a range 

 2Trll in x* 



Now, a change in the energy within a given space may be caused either 

 by the energies of the entering fluid and of that which is flowing out being 

 different, or by the rate at which the boundary-pressures do work on the 

 contained fluid being other than zero. In a length 27r// the first cause is 

 ineffective, as the velocities at the two ends are identical in value ; and as the 



* As far as x is involved, some of the second order terms in u are constant, and others have a 

 period irjl; ^udx will vanish only it the former terms are annulled, which may of course be 

 done by suitable end- conditions, as any function of y can be added to u. ■ ' 



