1917] and
AT THE SURFACE OF DEEP WATER
485
B ( U2 - 2gy) = 2a (8/3 - Sg) cos x + 2S/3 cos 20
+ 2 (287 - «S/9) cos 3as - 6a8y cos 4«................(25)
For the present purpose we need only to introduce S/3, and with sufficient accuracy we may take
8 ( U2 - 2gy) = 2S/3 cos 2a;......................(26)
We suppose S/3 = - '000,000,2, so that the new value of JB is - '000,052,6. Introducing corrections according to (26) and writing only the last two figures, we obtain column 5 of Table I, in which the greatest discrepancy is reduced from 10 to 4—almost as far as the arithmetic allows—and becomes but one-millionth of the statical difference between crest and trough. This is the degree of accuracy attained when we take simply
•\^ = y — ae~y cos DC — fte~-y cos 2# — ye~3v cos 3&,............(27)
with a = -3^, ff and 7 determined by Stokes' method, and ft determined so as to give the best agreement*.
[1919. Eeference may be made to Wilton, Phil Mag. Vol. 27, p. 385, 1914; also to Havelock, Roy. Soc. Proc., Vol. A 95, p. 38, 1918.]
[* If we include the first 3 terms of (25), and write
S (t/2 - Zgy) = -000,000,2 cos x - -000,000,4 cos2x + -000,000,2 cos3z,
corresponding to 5/3= --000,000,2, 5y= + -000,000,04, Sg= --000,001,2, we find that the corrected values of the last two figures of U2-2gy-l, given in footnote J on p. 483, become 45, 45, 44, 45, 46, 46, 45, 46, 45, taken in the same order; these results would not be affected by including the term in (25) involving cos 4#. Thus the greatest discrepancy is reduced from 11 to 2, becoming only half one-millionth of the statical difference. The new values of /3, 7, and g, thus determined so as to give the best agreement, are 8= - -000,052,6, 7= -000,000,94, g = -989,736,2. W. P. S.],1 •012,271,4 . 167,6