to deduce forms of minimum wave resistance in case such exist. One must, of course, 

 take care in formulating such a problem in order that a solution will exist. For 

 example, it is not sufficient to fix just the volume and ask for a form of minimum 

 resistance, for by distributing the volume deeper and deeper the resistance becomes less 

 and less without reaching a minimum. A solution, if it exists, will of course depend 

 upon the Froude number. This problem has been treated extensively by Weinblum 

 and also by Pavlenko [1] and Sretenskii [1]. 



If one fixes the centerplane section of the ship and its volume, and uses 

 Michell's integral for the resistance, the mathematical problem is to minimize a 

 quadratic form subject to a linear constraint. This leads to the following integral 

 equation for £ x (x,y) : 



SSdtdv UZ,v)K(f(x - £), f(y + ,)) = kx, (x,y) in S . (69) 



A solution £ x is sought such that 



ffdtdvXz&v) = V, 



V = fixed volume. 



(70) 



The latter condition will determine the constant k in the integral equation. Other con- 

 straints could replace this one, or be added to it. For example, one could fix £ x (Vi,6)., 

 £(0,v), etc. The integral equation is of the first kind, and these are notoriously difficult. 

 Sretenskii [1] deals with the analogous problem for the infinite vertical strut 

 (although he states that his result is valid for "elementary" ship), and claims to prove 



F = 0.3 2 5 



F= 0.340 



F = 0.355 



F= 0.385 



F= 0.430 



Figure 4. 



124 



