308 Mr Priestley, On the Oscillations of Superposed Fluids. 



(5) and (7) shew that F{t), A^, B^ are not changed by this 

 closer approximation. 

 If we write 



4i= p/3^~i sinj9^ + Ci, 

 B^= — p^k~^ sin pt — A> 

 we obtain, from (6), 



G^-D^ = - ^pk^' (p, - p^)/(p, + p,) . (sin pt + sin Spt). 

 The part of Gi and D^ depending on sin pt is indeterminate as 

 any addition to that part simply means a change in the amplitude 

 of the principal oscillation. 

 We take 



C\ = ^pk^^pj(pi + p.2) . sin pt + C^ sin Spt, 

 Di = ^pk/3^pi/{pi -r P2) ■ sin pt + A sin 3^^. 

 If we use these values in the two equations (6) and equate the 

 coefficients of sin pt and sin Spt to zero we have four equations. 

 Two of these are the same and lead to the new period equation, 



p^ =gk{l- s)l{\ + 5) . [1 - P^yg^ (1 + sO/(l + sf] (9), 



the other two give equations for G.^ and Do, the solutions of 

 which are 



C^2 = - i-^pk^' (5pi^ + pi - 12p,p,)/{p, + p,y . sin 3^^, 

 A = - ^pkl3' {pi + 5pi - 12p,p,)l(p, + p,y . sin Spt 

 Equations (8) give 

 As/pk^^ = ip^ (3p2 - pi)l(pi + p^y sin Spt 



+ T6P2 (9p2 - 5pi)/(/Ji + pif sin pt, 

 Bs/pk^^ = - ipi (3/Ji - p2)/(pi + pif sin Spt 



- 16 Pi i^Pi - ^p2)/(pi + P2)' sin pt. 

 If we now denote the amplitude of the principal oscillation 

 by h we find for the equation of the wave profile 



y = [b cos pt — ■^k'^lf (pi + pi)l{pi + p^y . cos Spt] cos kx 



kb^ 

 + — (/?!- p2)/(pi + P2) cos^ pt cos 2kx 



+ ^k""})^ (/3i - S/J2) {Spx - p2)Kpi + pif cos^ pt cos Skx . . .(10). 



Comparison with propagated wave. 



Wave Profile. 



Putting ti = in equation (6) of section I we obtain, as the 

 equation of the profile of the propagated wave when the upper 

 liquid has no stream velocity 



y = h cos kx + ^kb^ (p^ — p2)/(pi + P2) • cos 2kx 



+ %k''h' {pi + pi - ^pipi)l{pi + pif cos Skx. 



