38 



The longitudinal function S(y) can be easily computed from tabulated 

 intermediate functions M(y) for a given X(^). 



Basic families of S(y) or S^(r) functions are plotted in the same 

 way as families of ship lines, with which they are associated, for instance by 

 keeping t = const and varying <t> by steps of 0.02. (Figures 24 to 28) S(y) is 

 a linear function of ^, t, when the associated ship line (water line) is also 

 a linear function of <6, t. Corresponding to difference curves for the ordi- 

 nates (n n^ri^; 0.1, 0) and (n^n^ng; 0; 1), difference curves of S(r) can be 

 plotted which we denote by A^S(r) and AjS(y)* These permit a whole set of 

 S(y) curves to be developed for various ^ and t values, when one curve S(7') 

 for , t is known; they are also a valuable help in various discussions. 



Using the squared values S^(r), the resistance of different ship 

 lines can be compared as follows for a given Froude number F: 



a. Calculate y = — 2 • O^^ly S^(y) values to the right of y^ are to be 

 considered for a given F. 



b. The area enclosed between the S^(y) curve (multiplied by the mono- 

 tonic decreasing function <j>{y) f( y ), the ordinate at y^ and the y-axis is 

 proportional to the resistance. Taking into account all conditions, generally 

 the first waves of the S^(y) are decisive for the determination of resistance. 

 For a first orientation, comparisons can be made without multiplying S^(y) by 

 <i>Hy) f(y). 



c. Values of y for which S(y) or S^(y) are close to zero characterize 

 the position of a hollow in the resistance curve , a y value just to the left 

 of a crest of the S(y) or S^(y) lines, corresponds to a hump. Thus we can 

 estimate regions of low and high resistance by simple inspection. 



Results of Investigations on Longitudinal Distribution of Displace- 

 ment : D W. Taylor's experiments have revealed fundamental relations between 

 the resistance and the standard parameters <j), t .** It has been shown that 

 theory has succeeded in obtaining results which agree closely with these ex- 

 perimental data. The following discussions will, therefore, deal preferably 

 with more refined form effects which are not so universally known. The suc- 

 cess of these Investigations depends to a high degree upon the use of mathe- 

 matical lines. 



*Some examples of these curves can be derived from Figures 2k and 25. 

 ♦♦Exceptions will be mentioned later. 



