Beasho 



I = L (4;o»4'o) - L (ij; - 4^0,$ - $^ 



■•I, 



in, - Mv - ^o^v) dS. (C. 8) 



S 



Is equivalent to the boundary value problenn for \\i. Here, the 

 boundary values, 4^0 a^^d ij;©* are given by (Co 3). Since a stream 

 function has an arbitrary constant, we should also consider the 

 modified problem with boundary conditions 



^o ~ " ^0 ~ ^* constant on S, (C.9) 



which Is the homogeneous problem. [ 22] 



If condition (C, 9) holds, the surface elevation at the fore and 

 aft ends Is C (Instead of zero for the condition (C.3)), but the x- 

 component of the velocity at the same points Is -(1 +gc), by (C.2). 

 Hence, the water flows In and out the body unless C = - l/g. Thus 

 an adequate condition for a surface piercing body Is 



4; = . z on S. (C.IO) 



Throughout this section, we have treated a class of functions 

 \\i and ^ which are finite and continuous everywhere. As long as 

 the Integrals considered exist, the method may be applied with some 

 minor changes to other classes of functions. 



The question of the uniqueness of solutions will be left to the 

 future • 



572 



