Mr Priestley, On the Oscillations of Superposed Fluids. 303 



For tbis case the correction to the wave profile has its least 

 value. (Note. The correction in the cos ^kx term has not its 

 numerically smallest value.) The wave form is given by 



y = h cos kx — r^ k%^ cos ^kx. 



[This wave is traced for the value kh = '5.] 



Thus waves travelling with velocity given by (7) are least 

 affected by the small terms. The corresponding stream velocities 

 in the upper liquid are given by (8). 



Fig. 1. 



c = [i^fc-i{i-s)]i 



A, wave form as given by first approximation. 



B, wave form corrected for terms of order /c^ft*. 



Stability of waves for a given value of u. The period equa- 

 tion is 



Pi U,' + P. Ui = gk-' (p, - p,) + k'^^ ip,' U,^ + pi U,')l(p, U,' + p, Vi). 



In the small term we can take the first order values of c and 

 hence that term may be considered a known function of k and u. 

 The period equation therefore gives as a quadratic equation for c 



c' (pi + P2) - 2cM/), + p-fw" = gk-^ (p, - p^) + P^2^2, 



where P = {p,^ U,^ + p.^ U,%p, U,^ + p, U,% 



