81 



The double sign appears because the hull consists of two essentially symmetri- 

 cal halves. In most cases it is sufficient to consider 



y = +y(x,z) [la] 



and to double the results. 



When dealing with resistance problems it is advantageous to let the 

 X,Y plane coincide with the free surface and the Y.Z plane with the midship 

 section. Calculations of buoyancy, however, are commonly performed starting 

 from the keel. Some differences in definitions and denotations arise because 

 of the two systems mentioned, but in our present calculations we use only the 

 coordinates in accordance with Figure 1 . The basic elements of the form can 

 be expressed as follows: 



1. Load water line y(x,0) = X[x] [2] 



2. Midship section y(0,Z) = Z [z] [3] 

 3. Longitudinal section y(x,z) =0 z = klx] [4] 



U. Sectional-area curve A [x] = 2j y(x,z)dz [5] 







for a rectangular center-plane contour 



H 



A[x] = 2 J y(x,z)dz ■ [6] 







Brackets [ ] are used to distinguish operations performed on dimen- 

 sional coordinates. The volume is given by 



+ « k[x] +1 



V = 2 J J ydx dz = J A[x]dx [7] 



-I -/ 



Dimensionless coordinates are introduced: '^ 



^ = x// 77 = 1 f=f [8] 



Thus 



y = y(x,z) = br?(^f) [9] 



and the surface is described by its nondimenslonal shape, 



a principal dimension and the ratio B/L and H/L. The nondimenslonal basic 

 curves are defined as follows: 



