Eggers 



and its x derivative are uniformly bounded for x < x^ — and to prove this for not 

 too peculiar ^^ ^^ should be possible with moderate effort —then we may drop 

 the contribution of S^ as well as x^ tending to infinity and may finally write 



0(2), ^(2), ^(2)^ 



^1 



<2)(X,Y,Z) = _2 

 4 



^11 



F(^.0 



^x'^^-^F.O G^(^,0,O + ^z'\^,eF,0 G^(^,0,O 



dfd^ 



4^Xo 1, 



F(^,0) 





G(f,0,0) d^ 



(13) 



and 



( 2) 



02 '(X,Y,Z) 



4777o 



.(2)(^,^) Gd^ dr;. 



(14) 



The second-order potential is thus produced by: 



1. A distribution of doublets with moments corresponding to the deviation 

 of the local first-order flow relative to the ship from a uniform parallel flow 

 (giving rise to the Michell distribution) times the local volume of the ship. 



2. A distribution of sources over the plane z= whose density is essentially 

 the time derivative in an inertial system of dynamic pressure (save a contribu- 

 tion of the local flow components in the vicinity of the ship). 



3. A line distribution of sources around the ship's load water line of output 

 corresponding to the local breadth times the wave slope in the X direction along 

 the ship's contour according to linear theory. 



One should observe that the potential appears only in derivative form. On 

 the other hand, no differentiability of the hull surface function F is required to 

 make the expression for 0' ^' meaningful. 



For numerical evaluation, the following approximations are made: (a) cal- 

 culating the potential from a distribution &(X,Y) extending over the entire undis- 

 turbed free surface Z = including the waterplane area, and (b) inserting the 

 flow components calculated for the plane v= rather than on the hull surface. 

 Both these steps require continuation of the flow potential and some of its de- 

 rivatives into the domain occupied by the ship. This is achieved by extension of 

 the corresponding Fourier series in v- Insofar as these series would not con- 

 verge for 77 = in the usual sense (for example, the series for ^y) we treat them 

 as generalized functions. The second term of 5(X,Y) in Eq. (12) will in general 

 become singular at bow and stern; however, for a symmetric hull we will find 

 that it can be left out for calculation of wave resistance. 



The error involved with these modifications will in general be of higher 

 order in e than the terms to be determined; nevertheless, it should be checked 

 for components not uniformly bounded in the extended domain. It should be 



656 



