304< Mr Priestley, On the Oscillations of Superposed Fluids. 



whence 



c = up-Api + p2) ^ 



± [gk-^ (pi - p,)/{p^ + p-d + PA-^- - it"p,p,l{p, + /3,)"']-. 

 Thus c is real if 



gk-' (1 - s)/(l + s) > »"5/(l + sf - PA:-/3-. 

 Hence the range of unstable wave lengths is smaller than that 

 given by the first order equation and the corrections for small 

 terms tend to make the sj'stem more stable. 



As in the first order case the shortest possible wave travels 

 with a velocity us/{l + s). 



II. Standing Waves. 



The method of the first part of the paper is unsuitable for the 

 discussion of standing waves so we proceed as follows : 



With the origin in the common surface and the y axis vertically 

 upwards we assume for the lower liquid 



(f), + if, = -ZA,„e-''''*^'=+'y\ 

 and for the upper 



The condition to be satisfied at the common surface is 



pi^i-gpiy-k piQi' = p-2<l>2- gp-2 y-h p-^qi - F (t), 



where 



qi" = k-'Zm-AJ'e''''^J + 2^--27wiJ.„i^„e(»^+»*' ^'J cos {m - n) hx 

 = k"Ai- (1 + '2ky) + 4^^^i-42 cos kx 



(to third power of the amplitude), 



and F (t) is a function of the time. 



This condition gives the equation of the common surface 

 Avhich we will write 



The condition that no part of either liquid shall floAV across the 

 common surface leads to the equations 



.(1), 



dt ^ dx ^ dy 



dU dU dU^^ 



dt ' dx '^ dy 

 where {uy, v^), (u^, v,,) are the x and y components of the velocities 

 in the two liquids. 



We proceed to find the values of the ^'s and B's by successive 

 approximations from the conditions (1). 



