308 DEEP WATER WAVES, PBOGRESSIVE OB [393
iri which I, V, c, c', are small compared with a, a'. Thus (6) gives
, . , ( d*a . d2a' \ dz{3 . 0 (1 — a cos so — « sm x)( — r^ sin % + -,— cos x — -^ sin J# v / \ at2 at2 J at*
, . N /da. do.' . \ r/, / • \
- (a cos a 4- a sm a?) ( -^ cos as + -?r sm x ] — {(1 - a cos a? — a sm a?)
\ flC £66 /
x (a sin so — a cos x) + 2/3 sin 2fc — 2/3' cos 2a + 87 sin 3« — 87' cos 3
x -r j
dt dt
This equation is to hold good to the second order for all values of x, and therefore for each Fourier component separately. The terms in sin* and cos x give
* * A /fu
0 ...................... (9)
The term in sin 2as gives and, similarly, that in cos 2$ gives
In like manner
and so on. These are the results of the surface condition Dp/Dt = 0. From the other surface condition (p = 0) we find in the same way
'
, d{3 a dot.' a da.
From equations (9) to (16) we see that a, a' satisfy the same equations (9) as do a, a', and also that c, c' satisfy the same equations (12) as do 7, 7'; but that b, V are not quite so simply related to /3, /3'.. (6)