72 



The most general solution consists of a distribution of singulari- 

 ties over the ship surface itself. 



For this case a resistance integral has been given by Havelock.^* 

 However, he himself adds somewhat resignedly: "it is not likely that it would 

 give any better agreement with experimental results; for the more we depart 

 from the simple narrow ship the more necessary it is to take into account the 

 effect of wave motion upon the distribution of fluid velocity around the ship." 



Thus three steps can be listed which may lead to a really compre- 

 hensive theory: 



1 . Determination of the distribution of surface singularities corres- 

 ponding to a given ship form. A solution by an integral equation has been 

 Indicated by Kotchine,^* and has been discussed for very low Froude numbers, 

 i.e., actually for a deeply submerged doubled body. Here, however, the most 

 important problem consists of finding the changes of singularities generating 

 a given form with speed (Proude number). No solutions so far are known except 

 an investigation of a submerged cylinder by Havelock.''^ It appears therefore 

 to be appropriate to start with the simplest bodies. 



2. The calculation of the real attitude of models due to changes in 

 hydrodynamic- pressure distribution and of the resulting resistance. A general 

 approach by Hamilton's integral is imaginable in principle, but nothing can 



be said as to the practical value of such an attempt. Probably it is more 

 reasonable trying to find corrections for simple bodies as under 1 . 



3 . The representation of the viscous flow by appropriate systems of 

 singularities as a base of resistance calculations. 



The difficulties in dealing with this general approach are very ser- 

 ious. Therefore, some simpler methods of improving Michell's integral have 

 been sought . 



Havelock recommended for high-speed vessels a method based on con- 

 centrated singularities; as originally applied by him, it means even a simpli- 

 fication of Michell's theory— the substitution of concentrated sources and 

 sinks along the center plane for plane surface singularities.*^ In so far as 

 this method leads only to simplified computations, we are not interested in it 

 from our present viewpoint; but it can be generalized by locating these 

 sources outside of the plane. 



Appropriate formulas have been developed by Lunde for this case.'^^ 

 When applied to a model investigated by Wigley, Lunde 's calculations showed 

 a somewhat better agreement with experiment than computations based on 

 Michell's integral. It is expected that Lunde 's method will be a useful means 



