Minimum Wave Resistance for Dipole Distributions lll 
a range of Froude number up to about 0.4, owing to the lower efficiency in wavemaking of 
the afterbody, which evidently causes an advantage when displacement is moved from the 
forebody to the afterbody. At still higher speeds the best position tends to agree with the 
theoretical position at amidships. 
It may be of some interest to compare the results for struts of infinite draft found by 
Messrs. Kotik, Karp, and Lurye with some experimental and calculated results for bulbous 
bows I published in the Proceedings of the North-East Coast Institution of Engineers & 
Shipbuilders in 1936. These calculations were made for a spheroid added to a calculable 
ship form, and the experiments were made with a fairing between the spheroid and the form 
which would tend to diminish any additional form resistance. Also the change of form was 
hoped to be insufficient to change appreciably the frictional correction to the wave resist- 
ance. Under these precautions the calculated and experimental curves showed, for the best 
position and size of the bulb, a definite decrease in resistance over a range of Froude num- 
ber from 0.25 to 0.5. 
The conclusion of these comments is that, owing to the uncertainty of the application 
of such results as those in the papers under discussion, it is advisable that they should be 
checked by actual measurements before any practical use be made of them. 
G. Weinblum (Institut fur Schiffbau, Hamburg) 
May I express my sincerest thanks to the authors of this paper and of the preceding 
paper for the aesthetic pleasure presented to me. Some general remarks may be to the point: 
1. It is a well-known proposition that ship theory can become too difficult for naval 
architects. 
2. It is therefore advisable for naval architects to love mathematicians. 
3. This love, however, should not be unreciprocated. 
4. The present speaker feels much obliged that his mathematic colleagues have given 
such a kind credit to his earlier work. Obviously these attempts were formally rather poor; 
nonetheless, they embrace some physical and technical ideas which have proved to be fruit- 
ful in the further development. Besides the authors mentioned, Dr. Guilloton has contributed 
to the discussion of the problem which is thrilling both from the point of view of ship hydro- 
dynamics as well as mathematics. 
Twenty-five years ago a contribution was made by Prof. von Karman at the 4th Interna- 
tional Congress in Cambridge, England, in which he pointed out difficulties encountered 
when dealing with the exact solution of the minimum problem. Among other things, he found 
curves with infinite horns shown in the paper by Prof. Karp and others. This finding caused 
a lot of confusion in the professional world. Unfortunately, Prof. von Karman has forgotten 
to publish these interesting investigations. 
A decisive progress has been made in the meanwhile by distinguishing between the 
shape of the singularity distribution and the actual body form. Although this difference is 
well known and has been clearly illustrated by Havelock, e.g., in the case of the general 
ellipsoid, no use was made of this fact in many earlier publications on our subject. I sug- 
gested in vain a thesis on this topic some 12 years ago but it was not completed. So we are 
