83 



The position of a centroid x /l = ^^ is given by 



We define an "elementary ship" as a hull shape described by (1), the equation 



ri{^,i) = X(|)Z(f) [21] 



and (2) a rectangular center-plane contour. Dlmensionless surface ordinates 

 are obtained as products of the corresponding LWL and midship-section ordi- 

 nates. Elementary ships are characterized by the following properties men- 

 tioned earlier: 



(a) k*[^) = X(|) [22] 



i.e., the dlmensionless water line and sectional-area curves are identical; 

 hence 



(b) = a 6 = afi [23] 



(c) All sections are affine to the midship section. It is an advantage of 

 dlmensionless representation that integral properties like area coefficients 

 of curves, etc., are invariant to affine transformations. 



In principle any continuous ship surface can be expressed by a poly- 

 nomial. This follows from Weierstrass' theorem: A continuous function y(x,z) 

 within prescribed boundaries can be approximated with any desired degree of 

 accuracy by a polynomial in x,z. Thus 





n m 

 X z 



mn 



n.m 



[24] 



are general expressions for the ship surface. Instead of [2k] we choose the 

 special form 



' = ' -^»m«"«'" 125? 



*The minus sign before X has been introduced by analogy with Chapman's parabolas. From a mathe- 

 matical point of view the plus sign is preferable in all formulas like [25], [27], [23], etc. 



