Newman and Tuck 



Vossers, G., 1962, "Some Applications of the Slender Body Theory in Ship Hy- 

 drodynamics," Unpublished Thesis, Delft. 



Wehausen, J. V., and Laitone, E. V., 1961, "Handbuch der Physik," Vol. 9, Sur- 

 face Waves, Berlin: Springer -Ver lag. 



APPENDIX I 



SOME GEOMETRICAL IDENTITIES 



Gauss's theorem applied to a closed surface consisting of the immersed 

 hull surface S, together with the waterplane, indicates that 



j I p n dS = J J J V P dx dy dz - J J p k dx dy 



interior 

 of hull 



rater 

 p 1 an ( 



(Al.l) 



and 



JJprxndS = I j frxVpdxdydz - rrp(yi - x[) dxdy , (A1.2) 



for any sufficiently regular scalar p(x,y, z) , n being the outward unit normal 

 and r the position vector xi + yj^ + zk . The following identities are obtained from 

 the above for some simple special choices of the function p(x, y, z): 



Jjp(x)ndS = J dx p(x) [i(-S'(x)) +k(-B(x))] 

 JJp(x)rxndS = J dx p(x) i(xB(x) - ^jjzdydzj 

 JJp(x) zndS = Jdxp(x) i (- ^ Jj zdydzj + kS(x 



J -'^S(x) 



// 



z ^ dy d z 



I P(x) zrxndS = dx p(x 



jJp(x^yndS = J dx p(x) [jSCx^] 

 J J p(x)y rxndS ^ J dx p(x) i f - J J z dy dz - ^^ B^(x) 



+ k xS(x) + ^ J|y2dydz) 



(A1.3) 

 (A1.4) 

 (A1.5) 

 (A1.6) 

 (A1.7) 



(A1.8) 



For instance, to prove (A1.3) we note that 



154 



