(m^,m^,mg) = -(n-V) (xxV(})^) (23) 



where x is a position vector. 



In the slender-body theory, the length of the ship is assumed far larger than 

 the beam and draft. With this assumption, the components in Equations (21), (22), 

 and (23) reduce to 



(xxn) = (yn -zn -xn„, xn ) (24) 



^=- (-2 i +-3^)^*1 (25) 



(m,,m^,m,) = (-n2<i'-. +^^3^1 +71113-21112, -xm„+n_, xm -n ) (26) 



By substitution of Equation (17) into Equation (14) , the free-surface condition for 

 the potential, (^ . (j = 0,l, . . .7) , is given by 



2 - 2 



g ^. - oj ^ . + 2icoU ip. + U <p. =0 for z = (27) 



jz J ^jx jxx 



With the assumption of 



g (p. - 0)2 <P. = for z = (28) 



Equation (27) will be applied in the outer region where the three-dimensional so- 

 lution is expected and Equation (28) will be applied in the inner region. The two- 

 dimensional Green function which satisfies Equation (28) is given by Wehausen and 



9 

 Laitone for a single-hull body as 



00 

 G2^(y,z) = ^ PV J ^^^^ dk - ie^^ cos Ky 



kz 



(29) 



