with the boundary condition on the hull surface. We have omitted the term 



2 

 0(e ) in reducing the Laplace equation to the two-dimensional form. Then 



we can include the term 0(e) which may be the second order in the pertur- 

 bation expansion. Next the inner solution for the slender body can be 

 expressed by 



<J> N = (J) (2D) + §1 (x) + z g 2 (x) + y g 3 (x) (117) 



where (J) , Reference 20, is the two-dimensional solution of the boundary 



value problem, and g 1 (x) , g„(x), and g„(x) represent the indef initeness of 



the solution which may be determined by the matching procedure with the far 



field behavior of the fluid motion. In the above solution, we include 



2 

 terms up to the second order, but we have omitted terms of 0(e ) in the 



expansion of the free surface condition, so that the third and fourth terms 



on the right-hand side of the above expression must be deleted. Then we 



are obliged to take the expression 



j) N = (2D) + §1 (x) (118) 



The free surface condition in the far field, on the other hand, takes a 

 different form because the differentiation with respect to y or z does not 

 affect the order of magnitude. Therefore, the leading term of the equation 

 gives simply the linearized boundary condition as follows. 



2 

 +U lx") * +g ff =0 atz=0 ( 119 ) 



For the periodical motion with circular frequency w, we can write the 

 complex form as 



L = $ e iWt (120) 



46 



