■fcmn — ~ ~ C mn ( 



The general exterior theorem [6] of the prolate spheroid reads then 



D — A \ 



Pn m Kkl) „ \ (35) 



Qn m KZl) 



which is not confined to the case of axial symmetry. Hence we see that the remarks 

 above referring to the problem of testing are substantiated. Furthermore, numerical 

 examples of such spheroids compared with Weitbrecht's tests show excellent agreement. 

 Theorems for more general bodies have been found and are now in the process of 

 publication. In conclusion, we can say that the vital part of the interaction force or, 

 more generally, thrust deduction, namely the interaction potential, has been successfully 

 clarified and established. 



b. The topics and formulations discussed in a. are still confined to two restricted 

 cases: the propeller is representable by a singularity distribution and, second, the 

 hull form is supposed to be completely given. In practice, however, we like to know 

 after the stage of preliminary design what minor changes in the hull form and the 

 propeller characteristics could be effected in order to obtain an optimum combination 

 of hull and propeller in the sense that for a given constant input power a maximum 

 useful power can be obtained. This requirement was first stated by Dr. Lerbs [2] and 

 expressed in terms of the local thrust deduction coefficients and the local induced 

 velocity components. The formulation of this optimum problem whereby the average 

 thrust deduction coefficients and the average induced axial velocity components were 

 expressed in terms of the hull form parameters, and the disturbance velocity has been 

 given in Final Report I [3]. At that time the general method of a. was not known 

 and the sphere was used as a substitute. Nor has the thrust deduction formula been 

 in its general form, as it was first derived by R. Hunziker [7]. This problem can now 

 be analyzed where the mathematical work can be performed in a straightforward 

 manner. 



c. Finally, the general method applied recently by R. Hunziker (see Discussion 

 by Dr. Hunziker) has confirmed the assumption that the essential information neces- 

 sary for the thrust deduction coefficient is the disturbance potential of the body in the 

 presence of the propeller in an ideal fluid flow, an assumption which was adopted a 

 priori, without proof, by the discussors in their previous work. 



In conclusion, we can say that some of the criticism expressed by other dis- 

 cussors on the fruitful use of the propeller theory will definitely be removed if the 

 state of the art of wake-adapted propellers as already available is applied to its fullest 

 extent to deeply submerged vessels. 



REFERENCES 



1. Lerbs, H. W.: "On the development of the theory of marine propulsion". Symposium on 



Naval Hydrodynamics sponsored by O.N.R., National Academy of Sciences, and 

 National Research Council, Sept. 24 through 28, 1956. Washington, D. C. 



2. Lerbs, H. W.: "Moderately loaded propellers with a finite number of blades and an 



arbitrary distribution of circulation". Trans. S.NA.M.E. Vol. 60 p. 73 (1952). 



3. Martinek, J., Yeh, G. C. K., and Crawford, L.: "Final Report on Research and Investi- 



gation on Thrust Deduction". Office of Naval Research, Department of the Navy, 

 Washington, D. C. Contract NOnr 1117(00) Oct. 31, 1953, Reed Research Inc. 



4. Martinek, J., and Yeh, G. C. K.: "Final Report II on Theoretical Studies of Wake and 



Thrust Deduction (A Contribution to Potential Theory in Three Dimensions)". Con- 

 tract NOnr 1445(00) Bureau of Ships Fundamental Hydromechanics Research Pro- 

 gram (NS-715-102) June 30, 1955. Reed Research Inc. 



5. St. Denis, M.: "On Some Recent Advances in Ship Hydromechanics". Chesapeake Sec- 



tion of S.NA.M.E., Jan. 15, 1954. 



176 



