Okr — ^tahility or Instabilitij of Motions of a Perfect Liquid. 27 



solutions having wave-lengths 27r//, 27r/?i in the x and z directions. From 

 these we obtain 



f d r, d\ fdu dv \ fdu iiv \ 



(30) 



dv' 

 dy 



( d ^ d\ fdiD dv\ ^ dvj 

 and from these again, taken along with the equation of continuity, we obtain 



The most general integral of this is 



V'v = F{x-^yt,y,z), (32) 



where F is an arbitrary function. With the initial value of v, given by (24), 

 this becomes 



ys-y = - (P + m^ -'r %•) B cos I [x - f5yt) sin my cos nz ; (33) 



and a particular value of v satisfying this is given by 



2v' sin {Ix + (m - l(5t)y} sin {Ix - {m + l^it) y] .^ 



{I- + m} + 11?) B QOB nz P + (m - l(5tf + n~ l^ + {m + l^tf + ti^ ' ^ 



This, however, violates the conditions that v should vanish at the fixed planes 

 2/ = 0, y = &. We accordingly add to the value of v, as given by this equation, 

 another, v'\ satisfying the differential equation 



VV' = 0, (35) 



as well as the boundary conditions 



v" = - v\ when y = 0, y = h. 



This value of v' obviously is given by 



2v' sinh \h - sinh \{l)-y) sin Ix - sinh \y sin { Ix + {m - l[it) h } 



(l^ + m^ + n^)B cos nz /^ + (m - l^ty + n^ 



sinh X(b-y) sin Ix + sinh \y sin j Ix - (m + l(it)h] 



P + [m + l^ty + n- 



(36) 



in which, as in (28), X denotes ^/P + n"^. And the value v = v' + v'\ 

 obtained from (34) and (36), is identical with that given by (28). 



If the solution of the three-dimensioned problem is completed, the expres ■ 

 sions for ii, w involve a transcendental integral, and are somewhat longer than 

 that found for v. I accordingly return to the simpler case in which n is zero 



The initial values of u, v may now be written 



- riiB . , ] 



^''0 = — 7 — sm ix cos my | 



Vq = B cos Ix sin my 



W''] 



