306 Mr Priestley, On the Oscillations of Superposed Fluids. 



Equating the coefficients of the absolute term, cos kx and 

 cos 2kx to zero in these two equations, we have the following sets 

 of relations : 



F'(t)-{p,-p,)p'fi'sm2pt = (2), 



(from both equations), 



Piii - p^Bj +9 (pi- P2) kA^ = 0] 



/^\ 



PiAi-p.B^-g(pi- p,)kB^ = Ol 



p,A, - pA + 2i?^ (pi + p.^ A, = - p,p'^' sin 2pt] 



p^ A, - pA - 2p' (pi + P2) B, = p,p'/3'^ sin 2pt\ '^ ^' 



(on making use of the period equation to eliminate g). 



From (3) we see that Ai is still equal to — Bi] we shall still 

 give them the value p^k~^ ^inpt. The period equation will be the 

 same as before. 



From (4) A^ = - ^p./ipi + p^) p^' sin 2pt, 



-B2 = - ipi/(Pi + p2)p^' sin 2pt, 

 and from (2) 



F(t) = - Hpi - P2)f^' COS 2pt + C, 

 where (7 is a constant of integration. 



To keep the origin in the undisturbed surface we take (7 = 0. 

 The equation of the surface takes the form 



y = (3 cos pt cos kx + \k^^ (1 — s)/(l + s) cos^ pt cos 2kx. 

 Srd Approximation. 



9 (pi - p2)y = Pi4i (1 + % + i^'2/') cos kx 



— P2B1 (1 —ky + ^k^y'^) cos kx 

 + pi Jia (1 + '^ky) cos 2kx 



— p^B^i^. — ^ky) cos 2kx 

 + (piAs - P2-B3) cos Skx 



— ipiA;2 [^1^ (1 + 2ky) + 4^.1^2 cos kx] 

 + ^p2k^ W (1 - 2%) + 4<B,B^ cos kx] 

 + F(t). 



On using above values for y, A's and B's in the small terms 

 and writing 



X(t) = F{t) + UPi-P^)P'^'^^^^pt> 

 this becomes 

 9 (pi - P2) 2/ = % (0 + cos kx [pi^i - pa^i] + cos 2kx [p^A^ - p^B^ 

 + cos ^kx [pi J.3 — P2-B3] + 1- (pi — P2) p^^"" cos 2kx cos^ pt 



,cx<^ + (7p,^ + 7p,^ - 10p,p3) cos ptl 



4- COS Skx \^^P'' + ^^^' - ^"^P'P^^ '^' ^^^ 

 - 1 +(9p,^+ 9p,^-38p,p2)cos_p«. 



+ ^V^PW(P: + /^0 



