Maruo 



water plane vanishes at both ends is taken as the asymptotic form of the opti- 

 mum ship. Then the left hand side of the integral Eq. (50) is integrated by parts 



dX( X ' ) ( 1 ) 



; — ; — K (x-x)dx =k 



dx 



(52) 



.( 1) 



where K is an integral of Eq. (49) with respect to x. Integrating Eq. (52) 

 three times with respect to x, and taking account of the fact that dX(x')/dx' is 

 an odd function of x' and K(x - x') is symmetric, one obtains 



dX(x') „(4) ,. , ' 1 1 3 _, , ' 



—;-^ K (x-x)dx =— kx-^ + kx 



dx D 



where 



( 4 ) , 



K (x-x') 



/.CO 



Z(z)e"^^'^dz 



.(4) 



COS ['y/V(x-x') 



K^" ' has an asymptotic form when T tends to zero as follows: 



cos [7/V(x-x')] I Z(z)dz 



d\. 



J A^^ 



(4) , 



K (x-x') 



1, 



d\ 



y^^^ 



- Y„ [y(x-x'): 



Z(z) dz 



where Y^ is the Bessel function of the second kind. By putting 



A(x) = 2X(x) Z(z) dz 



JO 



that means the area of the transverse section, Eq. (53) becomes 



-.T A 



dA(x') 



1-^ Y^ [7(x-x')] dx' = -F-kx^ + k': 



27'' L ' ■ ] dx' 



This is an asymptotic form of the integral equation. If the condition 



dA(x) 



dx 







(53) 



(54) 



(55) 



(56) 



(57) 



(58) 



at X = t-t is employed, the end effect does not exist and the minimum wave re- 

 sistance is given by 



1032 



