and, within certain limits, of the vertical distribution of displacement. By 

 introducing additional functions we are able to represent changes in different 

 sections (for instance, inclination to the vertical, variation of fineness, 

 etc.) and waterlines.* 



Analytical representation of ship forms, developed for the purpose 

 of eliminating the mold loft, meets with two essential difficulties: 



1. The longitudinal profile is generally not a rectangle but a curve, 

 often with comers and discontinuities. This complication probably can be 

 overcome by a method already proposed. -"-"^ 



2. The representation of full sections. Here the use of high-degree 

 parabolas leads to regions with high curvature, which are detrimental from a 

 hydrodsmamic as well as a practical viewpoint. Some improvements can be made, 

 but as long as this difficulty is not overcome, it is not suggested that the 

 formulas be applied for construction. However, readers Interested in the prob- 

 lem may refer to a paper by Childsky,* where a criticism of the present meth- 

 ods and indications of some further development may be found. 



An Important question arises: Should the usual method of ship de- 

 sign, based on the sectional-area curve A(x), design waterline X(x), longi- 

 tudinal contour C(z) and midship section Z(z), be altered when using analyti- 

 cal expressions? Within the present range of application there seems to be 

 no need' for a radical change. Of course, the polynomial A(x) must have a suf- 

 ficiently high degree to comply with all functions of x representing the 

 ship's surface, and a further difficulty arises when the longitudinal contour 

 departs from a rectangle, because then the equation of the hull cannot be 

 easily expressed by a polynomial. 



The use of algebraically defined surfaces enables us to fit ship 

 forms in a rigorous manner, and especially to describe in a definite way 

 changes and variations in the forms. In applying the equations to problems of 

 resistance, motion in a seaway, etc., we expect to deduce results capable of 

 generalization and to establish parameters which are characteristic for the 

 problems Involved. An interesting, if not very important, application of 

 mathematical ship lines is the development of reliable approximate formulas 

 for ship design (position of centroids, moments of inertia) as already indi- 

 cated^""* and extended by Sparks.®^ 



*Examples of waterlines and sectional-area curves are given in Figures 15, l6, and 17 on pages 35, 56, 

 and i+1. 



