300 Mr Priestley, On the Oscillations of Superposed Fluids. 



the last summation being taken once for each pair of values 

 m and n, 



= U^-' [1 + A-^i- + 2k [A, cos A-^i + . . . + SA, cos SJc%] 



+ 2h-{2A,A,cosk%}l 

 neglecting terms of order higher than 1^(3'. 



Expanding ly'>S'i by the Binomial Theorem and neglecting 

 terms of order higher than k^^"^ we have 



IIS, = C^i- r 1 _ j^^^A^^ _ 2k [A, cos A-^i + 2^, cos 2A-^i 



+ 3^3 cos 3A;^i} 

 -2A--^li[2^oCosA-^i} 



+ 4^-2 [A;' cos^ A:^i + 4>A,A., cos A-^i cos 2A;^i} 

 + U-'A,^ cos k% - Sl^A,^ cos* A;^i 



We pass to the corresponding value of 1/aS^2 by changing the 

 sign of A; and using B coefficients. 



Since the origin is in the undisturbed surface the pressure 

 equation is satisfied over the line y — when there is no wave 

 motion. This gives 



p,U,' + 2gpA = P,W + ^gp-2G,. 



Using this condition we obtain the following relations from 

 the pressure equation. 



1st Appi'oximation. 



2gp,Ao = 2gp.B„ 

 which, with A^y — B^ (equations A), gives 



.4o = 0, 

 ^0 = 0. 

 2nd Approwimation. 

 Pi Ui' (- 2A;^i cos Ar^i) + 2gp, (A,, + A, cos A-^i) 



= p.^u/(2kB, cos k%) + 2gp^ (B, + B, cos A;^,)- 

 Remembering Ao = Bo, Ai = B, = ^, (B), and 



^1 = ^2, (A), 



this gives A,, = B^ = 0, 



and g (p, - p.^/k = p, LV + p-2 U.f, 



the ordinary period equation*. 



Srd Approximation, 

 pi Ut" (- 2A;^i cos A;^i - 4^vl2 cos 2A;^i - t-Ai' + ik^Ai" cos- A-^j) 



+ 2(7/3i [^0 + ^1 cos A;'^i + A. cos 2A-^i] 

 = p.,U.i {2kB, cos k% + 4<kB.2 cos 2k%, - k-B^^ + 4^•-5r cos^ k%) 



+ 2gp.^ [Bo + Bi cos k% + B^ cos 2A;^2]. 



* Lamb, Hydrodynamics, § 224, ed. 1895. 



