Current Progress in the Slender Body Theory 

 J J J Vp(x) dxdydz = i J dxp'(x) • J J dy dz 



n t e r 1 o r 



o f 



hul 1 



where S(x) is the area of the cross section at x and is for the last step assumed 

 to vanish at both ends of the ship. On the other hand 



-k J J p(x) dx dy = -k I dx p(x) • J dy 



water- length widthof 



plane of waterplane 



ship a t X 



= -k J dx p(x) B(x) , 



where B(x) is the waterplane bea;m at station x. The remaining identities (A1.4) 

 to (A1.8) may be proved similarly. 



The above identities are exact for a hull of arbitrary shape (providing it is 

 symmetrical with respect to y) and for an arbitrary function p(x). Their prin- 

 cipal use, of course, is for evaluating the forces and moments on a slender ship, 

 in which case p(x), zp(x), yp(x), will be identified as terms in a Taylor series 

 for the pressure on the hull. In addition, if the ship is slender, some of the 

 terms in (A 1.3) to (A 1.8) may be dropped to a consistent order of approximation 

 in e. For instance, in (A1.4) the term 



If J J ^^y^" 



is of order e^ whereas xB(x) is of order e; the former will be neglected when 

 (A1.4) is used in obtaining (2.21) and (3.32). Equations (A1.7) and (A1.8) are used 

 to obtain the sway, roll and yaw damping coefficients (2.14). 



It was not necessary to use explicit representations of the components of 

 the unit normal n in the above, but for later reference we now derive these using 

 a particular equation 



z = Z(x,y) 



describing the hull. Clearly then 



n = (iZ,+ iZy-k) [l^Z^+Z^] 



155 



