A Vortex Theory for the Maneuvering Ship 



4>'(0) 



ALUa 



- \j roj(77') x(7)')d77' + Jj Sy\(m,^co) n(m)i^ dS(m) 



+ JT ^ S7;(m,0) [z(m) n(m)i^-x(m) n(m)iJdS(m) 



> (38-) 



,'(+00) = 



and 



A')li;(t') . IpALU^a; [(J) ^ 0'(O) + j (J) ^ ^0'(t'-T')dr'l , (39) 



L t '^ 



This last formulae holds even when (w/U)^/ is discontinuous for t' >0. 



The expression of the deficiency A'!)ii{(t') and the expression of 1\i[ (t') 

 could be subject to comments similar to those made above about A'Zi(t'), 

 A'x;(t') and z[^^\t'), x[^'\t'). 



In the following paragraphs, the effects of z[ (f), x[ ^ (t') and 

 %.l^ ^\t') will be included in the contributions of the accelerations in the set of 

 forces due to the pressure p.. 



In the first draft of this paper, we gave an affirmative answer to the follow- 

 ing question: is the deficiency A'?j fixed with respect to the body? But in fact 

 the proof given was not valid. 



In the case of an airfoil of infinite aspect ratio, it is possible to show start- 

 ing either from the momentum of the fluid Q or from the pressure p'(m^,t'), 

 that the deficiency of the lift and its moment are actually proportional to one 

 another, the ratio being independent of the time.* We think that it is true also 

 in the present case. But the proof would require a finer analysis that the one 

 given above, although the latter is sufficient in order to yield the structure of 

 the main formulae. 



In return, however, we have, in cases "bg " and "b2," 



7o(m,t') - §{r,^(v') F*(t')\m} , yl(m,t') = §{r^ ^(v' ) F*( t' )\va} , (40) 



g being the linear and homogeneous functional defined by 



y\(m,t') = Pj{r^^(v') F*(t')|m} . 



For this reason, it seems that the three functions 0,0,0' introduced here are 

 suitable also in the cases "bg" and "bj" as in the case "bj." We will admit this 

 fact, at least for the sake of simplifying the writing. For instance, we will have: 



*See, respectively, (2) and (8). 



859 



