those of $. and ty . when the uniform flow has an opposite direction. 

 Then there are relations 



3$. 3$. 3iK 

 1 _ x r j 



3n 3n ' 3n 

 while the free surface conditions are 



3^ 

 3n~ 



* 2 * * 



2 A 8$ i 9 8 $ - 8< ^ 



- 0) <£>. - 2i0)U -^ + U ri + g v- 1 = 



i 3x . 2 ° 3z 



dx 



2 * 9^* o 9 2 ^* ^* 



- u> V - 2icoU 3- 1 + U — ^ + g -5-^ = 

 1 3x „ 2 ° 3z 



3x 



If we put, for simplicity, 



io>$ . + ii . = $ . , ico$ . + \b . = $ . 

 1 1 x 111 



the expression for forces and moments takes the form 



F. . = e 



p£ iJI *±£ 



dS 



(323) 



(324) 



(325) 



(326) 



We apply Green's theorem in the space bounded by S, Z n , I, and £„ such as 



U B 



illustrated in Figure 3. 



SI 



VV e+z b 



30. 3$. . 

 1 3n 3n j 



dS = 



If we consider the case of infinite depth, the integral on Z vanishes, 



B 



while the integral on I vanishes as the radius of the cylinder tends to 

 infinity. The integral on the horizontal plane is 



Jjy V I 



— $ . ) dxdy 



115 



