tensor m. . in general vanishes in the presence of a free surface and that the 

 conclusions of Kirchhoff concerning the influence of the symmetry of the body 

 form are no longer applicable. Again, we must be satisfied with an indication 

 only. 



As originally indicated, the hydrodynamic inertia quantities in our 

 case are functions of the body form and the density of the medium only. 



The coefficients k = m /m , etc., are pure numerical values, which 

 characterize a given geometrical configuration and which are roughly compara- 

 ble to the fineness ratios in the various directions. These physical quanti- 

 ties are easily determined, and in this lies the great simplicity of the con- 

 cept under consideration. 



Despite the abstractness of this theory, it yields valuable practi- 

 cal results in cases where the basic hypotheses are satisfied. Above all, 

 this is the case for accelerated motions of the body (ship) such as starting 

 conditions, oscillations, vibrations, and impact phenomena. 



However, even in calculating the moment M , which a body moving 

 with a rectilinear uniform motion at an angle of attack a experiences and 

 which can be calculated by the so-called Munk Formula 



M y =^>(k y - k x )¥ U 2 « [2] 



useful results are obtained if the circulation actually remains small as is 

 the case for a "regular" body of revolution. For such a body, experiments 

 give a "correction factor" of —0.85- For ship models with a sharp stern, 

 the author and others found a correction factor which naturally differed more 

 from unity and in special cases amounted to~0.6o. 



1 .3 The explicit calculation of apparent masses is not simple even in 

 the classical case. With regard to the two-dimensional problem, we refer to 

 the treatment of K. Wendel. 2 The three-dimensional problem has been solved 

 for the sphere and the ellipsoid; in addition, further results can be de- 

 rived from a theorem of Munk, 3 according to which the apparent mass, for 

 example, m' = m + m , of a body which is built up from a doublet distribu- 

 tion [a amounts to 



= nj>* &- 



With this relation, the values of m and m = m are given directly 



x y z ° J 



for a wide class of bodies of revolution. For the fundamental three- 

 dimensional shapes which in simple cases can be produced by distributions of 

 singularities in a plane (the so-called "Michell ship"), difficulty in con- 

 structing the body arises because no stream function exists. In fact, this 

 problem as yet has not been treated at all, although it is solvable in 

 principle. 



