298 Mr Priestley, On the Oscillations of Superposed Fluids. 



I. Propagated Waves. 



Reducing the problem to one of steady motion by the usual 

 artifice we express the coordinates of any point in terms of the 

 velocity potential and stream function by means of the equations 



for the lower liquid, and 



«t = 



for the upper. 



Ai, A^, Az ... etc., Bi, B^, B^... etc. are two series of quantities 

 in descending order of magnitude. 



The conditions to be satisfied at the common surface are 



Xi + iyi = x^ + Ly^ (3), 



and the pressure condition 



P,I8, + Igp, {y, + CO = p,IS, + ^gp, {y, + G,) (4), 



and Cj and Cg are constants. 



We have yet to choose our origin for -y^ and we select the 

 common surface as the line -^^r = 0. 



Writing '^ = (f)/U, condition (3) gives 







We proceed to find relations between the A's and B's from this 

 equation. 



1st Approximation. 



^2 "~ ^1 = * (-Do — -^o), 

 whence Aq = Bq] 



and ^1 = ^2] 



We take as the common first approximation to ^1 and ^2 the 

 mean of the two exact values which we denote by ^. 



2nd Approximation. 



^^ _ ^, = t {B, - Ao) + (A, + B,) sin A;^ + t{B^- A^) cos k% 

 whence 



^ = 5, = /3(say)' 

 ^2 - ^1 = 2/3 sin ^^ I ,g. 



.-. % = ^ + ^smk^f ^ ^' 



^^ = ^-^sinA;^ 



