Maruo 



any function can be expanded into Fourier series of ce^. As A(x) is an even 

 function of x, the expansion for a(0) contains odd terms only. 



o-(d) - ajcej(i9,q) + 83 063(19, q) + ajCe^C^.q) + ••• . 



There is also a relation 



L( 2n+l) 

 A2r+1 ce2„^i(9,q) 



where Aj^ + i is the Fourier coefficient of ce 2^+1, such as 



E( 2n+l ) 

 A^^^i COS (2r + l)5. 



n = 



Substituting these relations in Eq. (62), the following equation is obtained: 



00 a, , > 



EV-" / ( 2n+l ) ( 2n+ 1 )\ 



3an+ice2„,i(e,q)/\2„,l = ^ Ik^A^ + k ^ A3 J ce^^^,(0,q). 



n=0 n=0 



Therefore the unknown coefficients are determined as 



^,„., fk.A!*""'.k,A<."*"V (64) 



°2n+l ^2n+l l"^ 1 "1 



The condition (63) gives 



CO 



Z-i ^2n+l ^1 - 77 



n=0 



and together with the condition a(0) - o ai e = and -n the arbitrary constants 

 kj and k2 are determined. Since 



6k^ V 



k = - -^ Z(z)dz (66) 



the wave resistance is given by 



R. .!!if!!:r. (67) 



Necessary coefficients for the calculation of Mathieu functions have been given 

 by Bessho and the optimum forms of simple slender ships are calculated. Fig- 

 ure 2 gives the best curves of sectional area of the simple slender ship. It has 

 been found that the minimum wave resistance of the simple slender ship given 



1034 



