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Full text of "Scientific Papers - Vi"

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.
.    (jib
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
fSa'a-a^            (32)
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)
o                                           o
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 ;
so that •
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. '
mi                         #(
Thus      .             -J
and                                %2n above the mean level exceeds numerically the maximum depression below it.