96 
Proceedings of the Royal Society 
I a. Nodal Tesseral Oscillations. 
For nodal oscillations of the tesseral type we must have 
a = m J - 1 , /3 = n J - 1 where m and n are real, and by putting 
together properly the imaginary constituents we find 
h = C 
sin 
cos 
at 
sin 
cos 
sm 
mx ny 
cos * 
• ( 10 ), 
where m, n , a are connected by the equation 
+ n 
2 _ 
o- 2 - 4co 2 
gv 
• (in 
Finding the corresponding values of u and v, we see what the 
boundary conditions must be to allow these tesseral oscillations to 
exist in a sea of any shape. No hounding line can he drawn at 
every part of which the horizontal component velocity perpendicular 
to it is zero. Therefore to produce or permit oscillations of the 
simple harmonic type in respect to form, water must he forced in 
and drawn out alternately all round the boundary, or those parts of 
it (if not all) for which the horizontal component perpendicular to 
it is not zero. Hence the oscillations of water in a rotating rectan- 
gular trough are not of the simple harmonic type in respect to form, 
and the problem of finding them remains unsolved. 
If <o = 0, we fall on the well-known solution for waves in a non- 
rotating trough, which are of the simple harmonic type. 
I b. Waves or Oscillations in an endless Canal with straight 
parallel sides. 
For waves in a canal parallel to x, the solution is 
h = He ~ ty cos (mx — at) ..... (12), 
where l, m, a satisfy the equation 
m 2 - l 2 = 
a 2 — 4(o 2 
—gE>— 
• ( 13 ), 
in virtue of (9) or (11). 
