39 



lead to reasonable ship's forms. However, when we restrict ourselves to fam- 

 ilies of curves expressed by polynomials with few arbitrary parameters, we 

 really obtain an ordinary minimum problem and do not need to bother about the 

 difficulties connected with the application of the calculus of variations. 

 Take for instance the family (basic form) 



„ = i _ $« . a 2 (£ 2 - | 6 ) - a 4 (£ 4 - f 6 ) [34] 



with two arbitrary parameters. The wave resistance R is given as a second 

 degree function in a and a . 



2 4 



R = 4B a 2 + 4B a 2 + 8B a a + 24B a + 2MB a + B [35] 



22 2 44 4 24 24 22 44 



where 



B *= 7t\ - 6?H + 9&? 



22 11 15 55 



B = M7h -1271) + 9ft 



44 33 35 55 



B =2% - 3?n + 97h - 677) 



24 13 15 55 E 



35 



B = 77? - 3?7? 



2 15 55 



B = 27n - 3771 



4 35 55 



B n = 3697] 



55 



differentiating R partially with respect to a and a , one obtains the min- 

 imum conditions 



P- = B a +B a +3B =0 



OS. 22 2 24 4 2 



2 



JJ- = B a +B a +3B =0 



Ca 24 2 44 4 4 



4 



whence 



3[B B - B B ] 



a = 2 44 4 24 



2 B B - B 2 



22 44 24 



3[B B - B B ] 



a = 4 22 2 24 



4 B B - B 2 



22 44 24 



136] 



[371 



These equations lead to results which are not applicable to practice 



'o 

 0.125] 



when y = 1 and of restricted interest when y = 2 (f/L is assumed equal to 



