Smith 



and take cti(p) - f (p), we obtain a sequence of functions 



a.^j = f + \J Kjf + \2J Y^^f + ... + >,ij K.f . 



As i^oo^ this gives for cr(p) an infinite series in k with a radius of convergence 

 equal to the distance from the origin to the nearest eigenvalue \ = - 1 in the com- 

 plex \-plane. Since k = +i also lies on this circle of convergence, the conver- 

 gence of the sequence a- is not assured when k = +l; it converges conditionally, 

 if at all. 



The slow convergence of the iteration formula reported in the paper is attrib- 

 utable to this property. By analogy with the Gershgorin integral equation for con- 

 formal mapping, in which the identical properties occur, I believe, however, that 

 the functions ?■ = 1/2 (o^i+j + o-.^), or some modified version of successive pairs 

 of approximations, may converge much more rapidly, and I would suggest that 

 such a modification be tried by the author. 



DISCUSSION 



L. Mazarredo 



Asociacion de Investigacion de la Construccion Naval 



Madrid, Spain 



I want to ask whether an analytical solution has not been attempted for three- 

 dimensional bodies. Although not essentially needed, it might be of great help in 

 obtaining an efficient, rapid, and accurate way of preparing formal data. We found 

 this out when we began to work on the potential flow around a three-dimensional 

 body that approximated a ship. Our approach was based on the classical relaxation 

 method, but this makes no difference. 



In this case, an analytical definition might help in finding the boundary con- 

 dition. Since the speed induced by a source (q) on a point (p) varies as (pq")"^, 

 an element may be replaced by any other, provided it is parallel, has the same 

 solid angle - as seen from p- and the same intensity of the original one. The 

 original element can also be approximated by a spherical surface element inside 

 the same solid angle and center in p, if we increase its intensity in order to 

 maintain crS. 



If we know the equations for the boundary lines of the original elements, a 

 change of coordinates in order to move the origin to p and the z axis-to-the- 

 normal to Sp would not present any difficulty. The unit vectors of these curves 

 would give spherical elements whose projections on the xy plane, when multiplied 

 by a(q)/cos qr !, will give the normal component of the speed. Thus, the in- 

 tegrations are reduced to surfaces on a plane. Of course, this idea, which is 

 very similar to one currently used in radiation transfer, would require small 



364 



