Maruo 



is called the effective area which is the product of the length and the mean girth. 

 Let us consider a dual condition of constant displacement and constant effective 

 area. Let us start with the formula for the wave resistance of a ship of length 

 2-1 and draft T as 



R=2p\]^y'^ II f('^'Z) fCx'.z') K(z+ z', X- x') dxdx' dzdz' (32) 



where 



K(z+z', x-x') = if e-y>-'(^^^') CP3 [yK(x-x')] ^'^^ (33) 



which is obtained by the integration by parts of Michell's integral. According to 

 the principle of the calculus of variations, the minimization of the wave resist- 

 ance under the conditions of constant volume of displacement 



V = 2 f(x, z) dx dz = constant (34) 



and of the constant effective area 



J-lJo 



f(x',z') K(z+z', x-x')dx'dz' = kj + k^ y- ^^ (36) 



S^ = 2\\ yi+J^y dxdz = constant ^^^^ 



gives a non-linear integro-differential equation as 



Bf 



3z 



where kj and k^ are constants. Integrating with respect to z, one obtains 



r'' r^ 1 if 



f(x',z') K ^(z+z', x-x')dx'dz' = kjZ + k^ - + cPjCx) (37) 



where 



K^'Vz+z', x-x') = - —f e-^^'(^^=^') cos [7\(x-x')]^!£L (38) 



and cpjCx) is an arbitrary function of x only. Since the Eq. (37) is non-linear, 

 an iterative method is employed. In the first place, the vertical gradient of the 

 surface 3f/Bz is assumed as small. Then a linearization of the integro- 

 differential equation is made by the exclusion of the non-linear term (3f/3z)^. 



1028 



