in an Annular Trough. 201 
The second equation provides the condition for the free surface 
which is 
® 
when z = 0. (See Lamb, Hydrodynamics, p. 371.) 
There remain the boundary conditions, namely, 
^ = 0, when z = — li, the depth (4), 
^ = 0, all round the boundaries (5). 
du w 
As we are dealing with cylindrical boundaries, it will be con- 
venient to express equation (1) in cylindrical coordinates: thus 
?» + + ( 6 ) 
Sr 2 rdr r> W l dz 2 w 
The equations (3) and (6) and the conditions (4) and (5) are 
all satisfied by 
<f) = {A J n (/ cr ) + BY n ( kv)\ sin n6 cosh k (z + h) cos mt. ..(7), 
if — m 2 cosh kU -f g/c sinh kIi = 0, 
or m 2 = gtc 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 ' ( tea ) + BY n ' ( ko ) = 0, 
AJ n '{tcb) + BY n ' Ub) = 0, 
or eliminating A and B, k must be such as to satisfy the equation 
W1-7TtI = 0 (8 a). 
j n (tea) J yi \/cb) 
If x = Ka , px = Kb and p = b/a, x must be a root of the equation 
Y n ( x ) Y n (px) _ ^ 
J n '(x) Jn'(px) " W 
Substituting for B, we get as a solution 
</> = A n \j n (Kr) Y (/cr)j sin nO . cosh k(z 4- h)cosmt ...(10), 
( * n \ K 0) ) 
