Chta-Shun Yth 
The equation corresponding to (89) is now, for terms of order k 
and free frgm =, 
vA 
+ ay oe 
2 i ( NEP a2? Yok? ovee ) Poranh oY 
0 
(97) 
f' (hibeceen da) =afi0 
sick Seam il aAK A "20" F 
where 
a 
= d_ gpl 
il 4f 08? a 405) 
0 
Differentiation of (97) gives 
{Po ; : 
+ + \_gp! z ee 
2( X98 P'g%a2 * Pov20 28 ?' yao) 6 Yoa) t (eg¥aq) - 
(98) 
Elimination of Vo, between (96) and (98) gives, after multiplica- 
tion by f, 
Lv = f oe Pahoa) ae enact N28 *'9)o +2 » (99) 
in which L is the operator on vgg in (9la). We see from the terms. 
of order x? in (96) that V4g Can be expressed in terms of vag, 
and is therefore known. Thus I in (98) is known, 
2 
* Equating terms of 0(x ) in (96) to zero also guarantees the ae 
tisfaction of the free surface boundary condition for terms of 0(k°x hy 
as can be seen from a Stieltjes integration of those terms at the free 
surface and from (82) . 
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