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stream-function in terms of x and y, and lately I have found that this i may be extended to give, as readily if perhaps less elegantly, all the of Stokes' Supplement.
Supposing for brevity that the wave-length is 2?r and the velo propagation unity, we take as the expression for the stream-function waves, reduced to rest,
^r—y —ae~y cos x — fie~~y cos 2as — ye~3v cos 3#,    ..........
in which as is measured horizontally and y vertically downwards. expression evidently satisfies the differential equation to which ^ is s whatever may be the values of the constants a, (3, y. From (1) we fm<
U* - 2gy = (d^idcc}* + (d^fdy)* - Igy       .
= 1 - 2-vJr + 2 (1 - g) y + 2/3e~w cos 2a? + 4>ye~3v cos 30
+ tfe-w + kfi~e-w + Qry*e-«y + 4,a/3e~*v cos x
+ Qayer*y cos 2» + l2/3ye~5y cos as .............................
The condition to be satisfied at a free surface is the constancy of (2).
The solution to a moderate degree of approximation (as already r to) may be obtained with omission of /? and 7 in (1), (2). Thus from get, determining ty so that the mean value of y is zero,
which is correct as far as a3 inclusive.
If we call the coefficient of cos x in (3) a, we may write with th< approximation
y — a cos x - | a2 cos %x -t- |as cos 3x .................
Again from (2) with omission of /3, 7, U* — 2gy — const. + 2 (1 — g — a2 — a4) y + a4 cos 2x — fa6 cos So;.
It appears from (5) that the surface condition may be satisfied with provided that a4 is neglected and that
In (6) a may be replaced by a, and the equation determines the ^ of propagation. To exhibit this we must restore generality by intro< of k (— 2?r/X) and c the velocity of propagation, hitherto treated as .Consideration of " dimensions " shows that (6) becomes
or                                                 c2 = g.
Formulae (4) and (8) are those given by Stokes in his first memoir.
By means of /3 and 7 the surface condition (2) can be satisfie inclusion of o4 and a5, and from. (5) we see that @ is of the order a4 aia well-known paper by Stokesf. In a supplement published in 1880J the same author treated the problem by another method in which the space coordinates x, y are regarded as functions of $, i/r the velocity and stream functions, and carried the approximation a stage further.