Recently, several fundamental investigations of forms for which the 

 sum k + k + k is to be a minimum have been undertaken. The sphere and the 

 circular cylinder play an important role here. However, these difficult 

 treatments contribute nothing at the present to the solution of our problems. 4 



1 .4 As is well known, existing results can be applied directly in ship 

 theory as follows: 2 We image the submerged portion of the ship about the 

 waterline and calculate the inertia coefficients of the double model in an 

 infinite medium. At the same time, a strip method is applied and a correction' 

 for the end flow is made on the basis of the results for the ellipsoid. In 

 many cases, one simply sets the values k , k , etc., equal to those of an 



^ y 



"equivalent" ellipsoid. 



The crude strip method frequently proves to be very good if the 

 motion treated is in a vertical plane, therefore, if the determination of k 

 and k is the primary concern. As we shall see later, the values of Lockwood 

 Taylor, Equation [12c], should then logically be used for motions in a hori- 

 zontal plane. This, however, is not always done. Physically, the described 

 procedure is naturally very unsatisfactory. 



However, here we must emphasize a difference between the physical 

 and the engineering considerations. Prom the latter standpoint, a special 

 accuracy in determining the added masses is not necessary in many cases, so 

 that one can be satisfied with approximate values. The problem of vibration, 

 whose solution requires, among other things, very accurate knowledge of the 

 hydrodynamic inertia values, constitutes an exception. 



In the next section, we shall make some observations on what has 

 been done and on what is being done at the present to arrive at a rational 

 procedure. 



2. FREE SURFACE 



2.1 When one can no longer consider the fluid to extend infinitely in 

 all directions, then it is obvious from physical considerations that the 

 hydrodynamic masses depend upon the existing boundary conditions in addition 

 to the body form. If, in particular, there is a free surface, then we must 

 take the formation of waves into account. In order to differentiate the 

 results in this case from our previous results, we shall provide the usual 

 symbols with wavy lines, for example, m , k , etc. 



First of all, consider a physical observation. It is clear that 

 the kinetic energy can no longer be represented in the form of Kirchhoff 

 because of the creation of waves. The kinetic energy, whose time rate appears 

 as generated power or, in the case of oscillations, as the damping effect, 



