Mr Priestley, On the Oscillations of Superposed Fluids. 299 



3r(^ Approximation. 

 %-X = i{B,-A,) + le-'^^ «i^ ^^ [B, e-'^^ - A , e'*^] 



+ I [B.,e-''''^ - A.e'^^^l 

 whence we obtain, by equating coefficients of different harmonic 

 terms in the imaginary part, 



B,-A,= W^] 



A,-B, = k^A (C), 



A, =B,] 



while the real part gives 



% = ^ + ^smk% + ^ (A, + B,) sin 2^^, 

 ^1 = ^ - ;8 sin ^^ - -I (^2 + B,) sin 2k^. 

 [/3 may now differ from its former value by a small term of 

 order A;/3^.] 



4<th Approximation. 



X-%=l{B,-Ao) + ie-'*[^ smk^+i(A,+B,) sm-2k^] [^^g-^fc^ _ A,&''^] 



+ i[Bse-''^^-A,e''^^]. 

 Equating coefficients of the different harmonic terms in the 

 imaginary part we obtain 



Bo-A, = kl3' ^ 



B,-A, = ^^k(A, + B,) 

 B,~A, = -k^' 

 B,-A,=^-^^k(A, + B,)) 



Pressure Condition. 



To determine completely the quantities A and B we require 

 another set of relations between them. These relations are found 

 from the pressure equation (4). To use this equation we must 

 find 1/^ in terms of sines and cosines of multiples of ^ and this 

 we proceed to do. 



Value of 1/^1. 



.(D). 



From (1) 



dzi 



dwi 



where , , , 



Wi = 9i 4- iyi. 



= - [(1 + Ml cos k% + 2k A^ cos 2k% + ...) 

 + L (Ml sin A;^i + 2k A^ sin 2^^i +...)], 

 when ^/tj = 0. 



Thus 8^ = Ur" [1 + 2klmA,n cos mk% + k^'S.m'^A,,,^ 

 + 2k^Xm7iAyiAn cos (m — n) k%], 



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