PART II. : HYDRODYNAMICAL THEORY OF TEMPERATURE OSCILLATIONS. 631 

 or, neglecting quantities of the second order, 



= ~W)^ {Am (6) 



§ 25. Neglecting vertical accelerations (i.e. assuming [ —) small, and the amplitude 



of the oscillation also small) and considering only horizontal acceleration, the differences 

 between the pressures on the two sides of the slices in their disturbed positions will 

 simply be due to the difference in density between the two liquids and the change 

 £ in the level of the interface, i.e. g(p — p')d% per unit area. 



§ 26. It is evident that the motion in the two liquids must always be opposite in 

 direction if the level of the free surface of the lake is to remain undisturbed. The 

 equation of motion for S' and S together is then 



V dx)A\x) dt* \ dx)A(x) dt 2 * vr H 'dx w 



If the two liquids are to remain in contact we must have 



d*{A'(x)£} = J^{A(x)J} 

 dt 2 ~ ¥ 2 ' 



— ffr-rtg ■ • ■ W 



so equation (7) may be written 



y{A( g )g> r p ( u\ p (,Jl 



df> LA(a:) V dxj A'(x) \ fa 

 Neglecting quantities of the second order and substituting the value for £ from (6), 



a*{A0r)fl ( P , P \ = a(o- P ')b(x) - 1 jr_L 3{A(sW ] (9) 



~W~ \A@) + AHx)l 9{P P){h b{x)dx[_b{x) dx J • {V 



say 



where 

 or 



where 



iife^r^-^ • • • ■ <u) 



u = A(x)£ and v =/dxb(x) 



1 A.(a:) + A'(a;)J 

 -rtp-pW>)gj ... . (13) 



P + .P 



A(x) A'(x) 



