. - - ^^ = 



dy 



Mr Priestley, On the Oscillations of Superposed Fluids. 307 



To apply surface conditions (1) we shall require the 

 ^i- velocities to the second order and the y- velocities to the third 

 order. 



These are given by 



Ui = — -— = kA^ sin kx (1 + ky) + 2k A^ sin 2kx 

 = kA-i^ sin kx 4- '2k A ^ sin 2kx 

 + kp^- sin kx cos kx ^mpt cos pt, 



kA^ cos kx {1 + ky + \k^y^) 



— 2k A^ cos 2kx (1 + 2ky) 



— SkAs cos Skx 



= — kAi cos kx — 2k A 2 cos 2kx — 3 A; J. 3 cos Skx 



— pk^^ sin ^^ cos pt cos" A;« 



— ^pk^^^/{pi + P2) • cos^^i sin_p^ [{^Ri — ^^Rz) cos ^a; 



+ (3pi- 17/32) cos 3Ar«], 

 with similar values for u^ and ■Vg. 



Writing these values in the surface conditions (1), using 

 known approximations to coefficients wherever possible and 

 eliminating g by means of the period equation in all terms 

 except those of the first order, we obtain the following sets of 

 relations : 



%'(0 = (5), 



piAi - p^B^ + 9 {pi- P2) kA^ 



"(40/31^ + 44/J2' - 132pi|02) sin 3pf 

 ' |_ + ( Spi^ + 1 2 pi - 4/9i Pa) sin pt_ 



Ri-Ai - p^B^ -gipi- P2) kB^ 



'(44pi2 + 40p/ - 132pip2) sin 3^^ 

 ' L + {12 p{- + Spi - 4|Oi/32) sin pt_ 



(6), 



p^A^ - p^B^ + 2^2 (pi + /92) ^2 = - p^y^'' sill 2^^) 

 Pii; - pA - 2p' (pi + p,) B, = /9i_p^/3^ sin 2pt] 

 PiA,-p,B, + Sp'{p^ + po)A, 



{21pi — 9^2) sin Spt 



^kp'^'Kpi + R2) 



= i,kfmp, + p,) 



•{n 



= - ^kp'^%/{p, + p,) 



pJ,-pA-^pHri + p2)B, 



= -^kp'^'p,l{p, + p,) 



_ +{5p^-9p2)sinpt 



"(21^2 — 9/Ji) sin Spt 

 + {5p2- 9 pi) sin pt 

 (8) 



