Then we have the relation 



2tt ? 



D = p I R d6 







I Kde I . 



)x 9R 2 



1 (V<j>) 2 cos 6 



dz 



+ P 



2tt 5 2tt 



f R d0 J || cos 6 dz - | pg f 



•'n n •' n 



C R cos 6 d9 (246) 



where the free surface elevation becomes 





(247) 



Thus, the lowest order terms are quadratic functions of the disturbance 

 velocity potential. A similar formulation is applied to the y-component 

 of the steady force, giving 



2tt 



F = p 



y 



R d6 



3<j> d± j 34 . 1 ,„,,2 , . ' . 

 ^"91" "3T + 2 (V<J)) +gz ^ sin 9 ' 



(248) 



On developing the formulation, a relation of the energy conservation is 

 utilized as well. The energy contained in the space bounded by the control 

 surface is 



E = p 



H 



~ (V<D) 2 +gz 



dV 



(249) 



If the control surface is moving with normal velocity v directed inward, 

 the rate of change of the energy is given by 



3 



£-fjJJW«.pJJ 



\ (V$) 2 +gz 



v dS 



(250) 



89 



