^(i-a^-q^-a/Xi-y 9 ; 



Figure 5. 



that no solution can exist among square-integrable functions. It seems to the author that 

 what Sretenskii takes as eigenfunctions of the kernel are not really such and that his 

 proof is not valid. Pavlenko also deals with the strut, side-steps the question of exist- 

 ence, integrates by parts with respect to x (remembering that £(±Vi) zz 0) and 

 treats numerically the resulting integral equation: 



IdSUZ)Yo(f\x -|j) = k, 



M < X, 



(71) 



where 



/ d£ £(!■) = A = fixed cross-section area, f(±}0 = 0. 



(7,2) 



Pavlenko replaces the integral equation by a set of linear algebraic equations and 

 solves these. Figure 4 shows the resulting strut cross-sections for several Froude num- 

 bers. It is somewhat amusing that for Froude number less than 0.325 Pavlenko got 

 negative ordinates near the ends. Weinblum LI] considered "elementary" ships repre- 

 sented by the product of two polynomials, so that his computed minimal forms are 

 for a very restricted class of functions. However, they show the same general behavior 

 as functions of Froude number as do Pavlenko's. Figure 5 shows the water-plane 

 sections for various Froude numbers. 



According to a recent paper by Dorr [1], the solution of the strut integral 

 equation, if it exists, can be expressed as a series of Mathieu functions ce 2n . In fact, 

 Dorr has shown that 



ce n ((3;f) = X n J da ce»(a)Fo(/| cosa — cos 01). (73) 



Then, if one expands the constant k in a series in ce„ (actually, only ceo,, are needed) 



k = ^j a n ce„(|8;/ 2 



(74) 



125 



