Mr Priestley, On the Oscillations of Superposed Fluids. 301 



Expressing the harmonic terms in terms of "^ instead of ^i and 

 ^2) and neglecting terms of order Jr/S^' we obtain, by equating 

 coefficients of different harmonic terms to zero, 



2 (p,Ao - p,B,) = - (pi + p,) kl3\ 

 9 (pi - P2)/k = pi U,' + p, Ui = K, (say), 

 p, U,' [3A;2/32 - 4>kA,] + 2gp, [A, - ^k/3'] 

 = p,U,' [2k'^ + ^kB,] + 2gp, [B, + ^k^']. 



These equations, combined with (C) above, give 



Bo = W [ m 



A, = kl3'p,U,'IK I ^ ^' 



B, = -k/3^p,U.//KJ 



We note that, as in the case of waves on a free surface, the 

 period equation is not changed at this stage of the approximation. 



4<th Approximation. 



In proceeding to the next approximation we adopt the same 

 method as before. The period equation is obtained from the 

 coefficient of cos k'^ in the reduced pressure condition. This 

 coefficient gives 



p, U,^ [- 2kA, -\k'^{A,+ B,) + i/c^yS^j 



+ 2gp, [A, + ik^ (A, + B,) - ^¥^' + k^A,] 

 = p,Ui [2kB, - \k^^ {A, + B,) - iP/3^] 



4- 2gp, [B, -ik^(A, + B,)- 1 A;^^^ - k^B,]. 

 Remembering that 



gk-'(pi-p2) = PiU^' + p,U,' 

 to the order k^ and 



B,-A, = ^^k (A, + B,) = Ik^^^ (p, U,^ - p, Ui)l{p, U,^ + p, Ui), 

 we obtain for the period equation 

 9k-'{p,-p,)-K=-k'^'{p,'U,* + piV,')l{p,U,' + pJIi)...{b). 

 Before discussing this result we proceed to find the wave 

 profile. To do this we require the values of the ^'s and jB's to 

 the order k"^^^. 



The coefficients of the absolute term and cos 2k'^ in the 

 pressure equation are the same as the con-esponding coefficients 

 in the third approximation. Hence A^, B^, A^, B^ have values 

 already given (E). 



A-i and Bg are found from the coefficient of cos Sk'^, which, 

 with the help of (D), gives 



^3 = 1 PiU,' i2p,Ui' - pJJi)K-^^'k\ 

 B,= lp,Ui{2p,Ui-p,U,-)K^^%\ 



