Mr Priestley, On the Oscillations of Superposed Fluids. 305 



1st Approximation. 



The equation of the free surface is 



(J (p, - p^)y=(piAi- P2B1) cos kx + F(t). 

 The origin being taken in the undisturbed surface, we have 

 F(t) = and 



g (pi -p2)y = (pi^i - P2-B1) cos kx. 

 The surface conditions give 



Pi 4.1 ~ P2^i + 9iPi- P2) kA^ = 0, 



PiAi - pA -9 {pi- P2) kB, = 0, 

 Avhence Ai = — Bi= p^k~'^ sin pt, (say), 



and p^ = gk{l — s)/(l + s), where s = p^jpi. 



Using the values of A^ and Bi, we have, as equation of the 

 surface 



y = ^ cos pt cos kx. 



2nd Approximation. 

 The equation of the free surface is 

 g {p-i — p2)y = Pi^i (1 + ky) cos kx — p^B^ (1 — ky) cos kx 



-!- P1A2 cos 2kx — P2B2 cos 2kx 

 -lp,k'A,' + ^P2k'B^ + F{t); 

 whence, using first approximations to Aj, B^ and y in the small 

 terms 

 g (pi - p2)y = (pi-^i - P2B1) cos kx + {P1A2 - P2B2) cos 2kx 



+ (pi — P2) p^^^ cos^ pt cos^ kx 

 - ^{pi-p2)p'/3' sin' pt + F{t). 

 The velocities, required for the surface conditions, are given by 



Wi = — ;^ = ^^1 sin kx, (to first order), 



ox 



Vi = — -^ = — kAi cos kx (1 + ky) — 2k A^ cos 2kx 



oy 



= — kAi cos kx — 2k A^ cos 2kx 



— pk^' sin pt cos pt cos^ A;a;, (to second order), 

 with similar values of U2 and v^. 

 The surface conditions give 



(/3j j-i — P2B1) cos A;i» + (pi^2 — ^2-^2) cos 2kx 



— (pi — P2) i'^/S^ sin 2pt cos^ ^ic 

 -HPi-P2)p^/3^sin2pi + i?"(0 



— (pi + P2) p^/3^ sin^ kx sin p^ cos^i 



+ (/ (pi — P2) [kAi cos Z?^ + 2kA2 cos 2A;a; 



+ A>p/3" cos kx COS pt sinp^] = 0, 

 with a similar equation from the other liquid. 



