Manchester Memoirs, Vol. xlvii. (1903), No. |,3. 9 



uniformly supported and the vertical through the centre 

 of gravity G. 



Ii = height of any point of the wave surface above 

 datum, at a distance x from G. 

 //o = mean height of wave surface. 

 A = length of wave from crest to crest. 

 Let 7 = height of G above datum, 



^= deflection of any point x in the centre line of the 

 vessel form the a.xis o{ x. 



The upward pressure on any element u{ length 8,r, 

 whose distance from the origin =,!', is 

 x = wB{/i -y + d)cx 

 .'. total upward pressure 

 +f 

 = / w£{/z -y + d)dx = W [displacement] 



if 7 is assumed constant, />., any small vertical motions 

 neglected. 



When \ = L this becomes 



or, finally, 



7vBd=7ii' \_y = h„] (A 



The equation expressing the equilibrium of the beam 

 at any point, neglecting the term involving the rotarv 

 inertia of the cross section, is 



JdF^^^^J^^-^"B{h-y^a)--zv 



= wBAsm-~{x + v/) (t)* 



*Equation 5 is only strictly correct as long as the angular displacement 

 in a vertical plane is zero. Since otherwise the upward pressure would 

 largely depend on the angle assumed by the boat at any instant, but as this 

 is small it has been assumed that the difference is negligible. 



