and lets 





(75) 



the solution of the integral equation is given by 



2 

 TO;/) = . <?(arc cos - 2x;f) 



VI - 4z 2 



(76) 



Unless G has compensating behavior at x = ±Vi, the solution has a singularity at these 

 points which deprives it of physical meaning; and, of course, the condition t,{±Vi) = 

 is not satisfied. This seems indeed to be the case for sufficiently small / (large Froude 

 number), either from a study of the asymptotic expansions of \ n and ce„ or from 

 the development of Y about the origin: 



(77) 



Y (x) 



2 x 

 -log- 



t 2 



1 - 



+ 



1 



+ 



2 T - 



(27 - 2) + 



The latter leads to the simpler integral equation for small / 



df f(f) log^Ja; -f| = b, 



\x\ < & 



log 5 = 7 = Euler's Constant, (78) 



and eventually to the solution 



fa) 



Wl 



(79) 



4a; 2 



independent of /. It is not known at present whether this behavior persists for small 

 Froude number. The matter should be investigated. The asymptotic expansion of 

 Michell's integral for small Froude number which was given earlier indicates that an 

 optimum form should have cusps at bow and stern and elsewhere avoid discontinuities 

 in slope and curvature. 



Establishing the existence of a solution in a physical problem may seem like a 

 luxury. However, unless a physically acceptable solution exists, Weinblum's investiga- 

 tions of optimal forms among restricted classes of functions such as polynomials lose 

 much of their significance; for the underlying assumption is that these are approxi- 

 mations to an exact solution. This applies even more strongly to Pavlenko's approach. 

 One must, of course, remain aware of the fact that ships are not necessarily "thin" 

 and that they do not sail in perfect fluids, so that the practical implications of such 

 studies are not necessarily important. However, they may put one in possession of the 

 techniques for finding, say, optimal forms for the forward half of a ship when the 

 after half is given. 



Several Ships and Ships Near Walls or in Canals. The wave-resistance 

 integral applies equally well to two or more ships in a row if they all travel at the same 

 speed. For let 



?(x,y) = fi(z,y) + Ux,y), (80) 



where ^(x.y) = except on S x and £ 2 (x,y) = except on S 2 , where the centerplane 



126 



