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Periodic Irrotational Waves of Finite Height. 42 
where : 
Ay = Dy + 8D: + 6D2+3(3D3+4Ds+...):-3 (Di + 2D2+ 3D3+...) cosh 2a, 
As = £D,+4D3+3(3D3+4Da+...)—3 (2D2+3D3+...) cosh 22, 
As = 4D.+4D3+3(4Di + 5D; +...)—3(3D3+4Ds+...) cosh 2a, 
A, = 12D34+7D.43(5Ds+...)—3(4Ds+ ...) cosh 2a, 
For the other side of equation (7), using I to denote the imaginary part Q 
of a complex quantity P+7Q, we have 
se = —Ie- (sinh? a +sin? p)%e~* 0-9/3 & Ber, (11) 
where Re® = cos ¢ sinh «—isin ¢ cosh a. 
To expand this in a form similar to (10), we notice that 
e- 10-43 = (1 — e720 g2ib 1/6 (1—e-2 e- 2) U6, 
(sinh? a+ sin? py? — ter el — e720 op) 2 (1 —e72a Cue) 2k 
Hence we have 
e = —I 4¢3 (sinh? «+sin? gd) 3 & B,(1—e-22e7)7/3 (1 — e728 e—2th)U/3 g2rid | 
r=0 
| (12) 
We now expand the two binomial factors after the sign of summation in 
series valid for the whole range of ¢@ and for all positive values of «. We 
can then write down the coefficient of ¢?”?, and so obtain 0f/dy, involving a 
series of sines of even multiples of @; as before, we take out a common 
factor sin d, and obtain the result 
= 1-4/8 sin g (sinh? @ + sin? )-9 (By cos 6-+ Bs cos3¢+...), (13) 
where the B’s are linear functions of the §’s, with coefficients which are 
functions of e~**. In practice, these can be obtained directly from (12) to 
any required degree of approximation; general expressions can be put in the 
following exact forms 
Bont1 = Boent1+ Bi Bi, 2ntit-.. + BrBrentit-.-., 
ine) oo 
Be 2n+1 = > C_25— > Cos, 
s=ntr+l] s=n—r+l 
eet (eye io — 
Ce ee 
—1(—141)...(—4+s—1) -2,.- 
Chas $a sal gts—1) ,-2%6 NeR(—145,—2,s+1e-"), 
Co = 3c F (—4, —2,1,e~*), (14) 
where F represents the hypergeometric series. 
136 
