Bessho 



T =iyjy [V4>]^ dT (3.2) 



and 



= 1^ 



t,^ dx dy (3.3) 



are the total kinetic and potential energies, respectively. L is just 

 the kinetic potential or Lagrangian. [ 5] Assume that the function 

 <^ has a given normal derivative 



^v- - ^v oil S and F. (3.4) 



Taking the variation of L, we have 



5L=-\\\ (t>V^6(f) dT +( ( <t>6(j)ydS + 



jy [<^6<j)^+{(iV(^)^- g;}6v] dS. 



Making use of (3.4), which is also true for the new deflected free 

 surface, we have 



6L=-yjy <t>V^6<l>dT + i^J p6v dS, [3,14] (3.5) 



where 



p/p = - *x- i(V4>f - g^. (3.6) 



Hence, if the pressure at the free surface vanishes, the 

 stationary value of L will be attained when 5<}> is harmonic. This 

 is just an extension of Kelvin's minimum energy principle, [ 1,4] 



On the other hand, if 6<f) is harmonic, then the stationary value 

 of L is attained when the free surface pressure is constant and 

 zero. The latter is an extension of Riabouchinsky's principle of 

 minimum added mass . [3,14] 



The variational problem can be transformed so that the con- 

 straint condition is converted to a natural condition. Let us add a 

 term which is zero at the stationary point. Consider the functional 



552 



