in an Annular Trough. 201 



The second equation provides the condition for the free surface 

 which is 



3k'g-° ^ 



when z = 0. (See Lamb, Hydrodynamics, p. 371.) 

 There remain the boundary conditions, namely, 



~ = 0, when z = — h, the depth (4), 



jS-0, aH round the boundaries (5). 



As we are dealing with cylindrical boundaries, it will be con- 

 venient to express equation (1) in cylindrical coordinates: thus 



d^ + id± + id^ + d^ = 



dr 2 r dr r 2 W 2 dz 2 



The equations (3) and (6) and the conditions (4) and (5) are 

 all satisfied by 



<f> = {A J n (icr) + BY n (fcr)\ sin nO cosh k (z + h) cos mt. . .(7), 



if — m 2 cosh ich + gic sinh kJi = 0, 



or m 2 = gic tanh kIi (8). 



If a is the inner radius of the trough and b the outer, then the 

 second boundary condition may be expressed by the equations 



AJ n ' (*q) + BYn (ko) = 0, 



'AJ n '( K b) + BYn'( K b) = 0, 



or eliminating A and B, k must be such as to satisfy the equation 

 Ynjica) Y n '( K b) ^ 



J n ' (KO) J n ' (Kb) 



If x = ica, px = K,b and p = b/a, x must be a root of the equation 



Y n \X) I n (pX) __ - ._. 



Jn{x) J n '{px) 



Substituting for B, we get as a solution 

 <£ = A n \j n (jcr) — ' l , .. Y(icr)\ sin nd .cosh k(z + h)cosmt ...(10), 



