Passing next to the condition p = 0, we see from (24). by considering the coefficients of sin x, cos as, that
~ ~Ji + 9a/ + terms of 3rd order = 0, clt
-7- + ga 4- terms of 3rd order = 0.
The coefficients of sin 2#, cos 2#, require, as in (14), (15), that
1 ,7/Q 1 /7/Q' r/2 _ nz
jf J. UiKJ , 7 J. U/AJ U/ It
b=--T7 aa, o = ---- ^r + ~~o g at g dt 2
Again, the coefficients of sin 3x, cos 3*, give
3a' ,0. q , ,,,
~ 8 (a ~da}) ........... >( }
c = - + |(a - a>) +
dt 2 8
8 Cda a) .......... ( }
When /3, /3', 7, 7', vanish, these results are much simplified. We have 6' = -att', & = |(a'2-tta), ..................... (33)
c' = _ 3a-' (a'a _ 3aĞ), c = _ ^ (3a'2 - a2) ............. (34)
If the principal terms represent a purely progressive wave, we may take, as in (17),
a = A cos nt, a' = A sin nt, ..................... (35)
where n is for the moment undetermined. Accordingly
6/ = -i^lasin2n*, -b = - ^A2 cos 2n£,
c' = M8 sin 87iĞ, c = | A3 cos 3?itf ;
y = A cos (a? - nt) - %AZ cos 2 (as - nt) + f J.8 cos 3 (x - nt), ...... (36)
representing a progressive wave of permanent type, as found by Stokes.
To determine n we utilize (28), (29), in the small terms of which we may take
Ğ = <7 a'dt = , cos nt, of = *- g I adt = sin nt,
so that * a?+az = A*n2. '
Thus . -J
and %2n above the mean level exceeds numerically the maximum depression below it.