Muvthy 



plane z = in the form 



<j>(x,y, f ;t) = 4>(x, y, 0;t) + f 





z=o 



Since f may be assumed to be of the same order as <i> , we may- 

 retain only the first term in the above expansion and write 



f = 



1 



1 / ,2 



t " V<l> x + 2 (V4>) 



+ 



P s (x,y;t) 



We therefore have 



JJ p s (f x t + r y i 



k ) dxdy 



and substituting for the derivations of f 

 1 



F = 

 P 



— (4> . - V$ + V$. V4> + P 



g xt xx x p s 



>* ♦ 



+ (* , - Vf + Vefc.V* +— - 



g yt xy y P 



s y ) j - k 



dxdy 



where the potential has now the argument (x, y, 0) and the integration 

 has thus been reduced to S Q which is the instantaneous position 

 of the part of the plane z = contained within the vertical projection 

 of the hemline of the skirts at the bow and the stern and between the 

 inner surfaces of the side hulls. 



The components of the pressure force are therefore 



and 



X 



S o ' 



V<t> + V*. V4> + P ) dxdy 



xx x P s 



x 



— MP (* l ~ v * + V* . V4> + — p ) dxdy 

 gJJ s yt xy y P s 



S o 



// 



p dxdy 



(II. 1) 



174 



