sources of strength e H over the hull surface H, the boundary condition is expressed 

 by the following integral equation: 



1 C C cos (r,n) 



e H (x h xi,x z ) -\ / / e H (ti,U,&) dH = q(xi,x 2 ,xi) (20) 



2*JJh r 2 



q(x h X2,Xs) = 2 



v s cos (n,Xi) + 

 1 C C ep 



d<j> P 



dn I H_ 



jp = - 



4tt / / r, Tp 



(21) 

 dF P (22) 



r = [<*i - ^ + (x t - £ 2 ) 2 + (x t - $,)*]» (23) 



r P = [(xi - x 1P y + (.r 2 - .r 2P ) 2 + (x 3 - x iP y-]i (24) 



By x the coordinates of the point of reference within the hull are designated, 



by | the running coordinates of the hull and by x P the coordinates 



of the propeller disc. 



The solution of this Fredholm equation of the second kind is known to be 

 feasible by means of the method of iteration of kernels. 



Having determined e H , the effective wake in the plane of the propeller may 

 be ascertained from the potential of the source distribution. The paper includes a 

 numerical example for a body of revolution which has been tested formerly by 

 Weitbrecht [22]. The thrust deduction coefficient obtained experimentally amounts to 

 0.128 and that by computation to 0.119. The difference lies in the right direction and 

 the order of magnitude of the differences is in agreement with the generally held 

 opinion, viz., that the frictional thrust deduction is a small quantity in the case of 

 rotational symmetry. 



In papers by Korvin-Kroukowsky [20] the interaction theory is developed on a 

 basis of the singularity concept in the form of a computational procedure. The 

 boundary condition is satisfied at a certain number of arbitrarily chosen control points 

 on the surface of the body which leads to a set of linear algebraic equations for the 

 strength of axially distributed sources and sinks. For the propeller, infinitely many 

 blades are assumed and the radial distribution of its bound circulation is replaced by 

 a rectangular one. Correspondingly, two free vortex sheets of different diameters 

 each consisting of a semi-infinite row of ring vortices are introduced. Their velocity 

 field is derived from Laplace's equation. Taking into account the velocities induced 

 from the propeller vortex system when satisfying the boundary condition at the con- 

 trol points on the hull modifies the strength of the body singularities. The modification 

 approximately corresponds to the image of the propeller system. From the modified 

 singularities of the body the effective wake follows. The thrust deduction force is 

 determined in two ways, vis., on the basis of Lagally's law from the perturbation 

 velocities of the propeller at the body sinks and, alternatively, using surface pressure 

 integration. Numerical results indicate that, for the example treated, the wake fraction 

 is not sensibly affected by the action of the propeller. These results show further that 

 the thrust deduction force decreases rapidly on the body when going forward from the 

 stern. The first four sinks nearest to the stern situated within 12% of the body length 

 already contribute 90% to the total force. 



As far as a non-viscous fluid is concerned singularity methods have proven 

 successful to understand the body-propeller interaction and to express this interaction 

 analytically. The results which are known today are restricted to axi-symmetrical flow. 

 The next step should be to explore the effect of a circumferentially non-uniform wake 

 on the interaction force. Replacing the propeller by a vortex system, additional vor- 



161 



