98 
Proceedings of the Royal Society 
If we give ends to the canal we fall upon the unsolved problem 
referred to above of tesseral oscillations. If instead of being 
rigorously straight we suppose the canal to be circular and endless, 
provided the breadth of the canal to be small in comparison with 
the radius of the circle, the solution (17) still holds. In this case, 
if c denote the circumference of the canal, we must have m = 
c 
where i is an integer. 
II. Oscillations and Waves in Circular Basin (Polar 
Coordinates). 
Let 
h = P cos (iO -crt) (18) 
be the solution for height, where P is a function of r. By (8') P 
must satisfy the equation 
d 2 P 1 df_ _ iff o- 2 - 4co 2 p 
dr 2 + r dr r 2 + gD 
and by (7') we find 
*= Aj- sin {i6 - - 2m: *'?) 
T = cos (ie - ^( 2w S - f) , 
This is the solution for water in a circular basin, with or without 
a central circular island. Let a be the radius of the basin, and if 
there be a central island let a' be its radius. The boundary condi- 
tions to be fulfilled are £ = 0, when r = a, and when r = a'. The 
ratio of one to the other of the two constants of integration of (19), 
and the speed a of the oscillation, are the two unknown quantities 
to be found by these two equations. The ratio of the constants is 
immediately eliminated, and the result is a transcendental equation 
for or. There is no difficulty, only a little labour, in thus finding as 
many as we please of the fundamental modes, and working out the 
whole motion of the system for each. The roots of this equation, 
which are found to be all real by the Pourier-Sturm-Liouville-theory, 
• ( 19 ) 
• ( 20 ) 
