where P is on SH⁽ⁱ⁾. Eq.(16) is the integral equation representing ship hull boundary condition with unknown variable of σ(i).

 We consider rewriting the forces expressed as eqs.(7). (8) and (9) to the forces defined at ship fixed coordinate system. Here, we denote the forces at the ship fixed coordinate corresponding to the 1st and 2nd integral terms in right hand side of eqs.(7), (8) and (9) as FAκ⁽ⁱ⁾ and FDκ⁽ⁱ⁾respectively. FAκ⁽ⁱ⁾ mean the time derivative of the velocity potential, and FDκ⁽ⁱ⁾ the quasi-steady pressure with respect to the square of the potential. By applying the following relations,

　

　

FAk⁽ⁱ⁾　is obtained as follows:

　

　

 In the equations, ^・ means the time derivative.

 We consider explicit expressions of the term which is proportional to the acceleration. Added mass and moment of inertia with respect to the j-th force induced by motion of the i-th mode for Ship k are defined as:

　

　

 Here, mji⁽¹⁾ = 0 for j ≧ 4 and mji⁽²⁾ = 0 for j ≦ 3, so that eqs.(17), (18) and (19) are rewritten as follows:

　

　

　

 Here, we omit the number of (1) and (2) at mji for distinguishing 2 ships because j ≦ 3 is for Ship 1 and j ≧ 4 is for Ship 2.

　

2.4 Motion Equations for Ship Maneuvering

　

 Motion equations of ship maneuvering (Euler's motion equation) for Ship 1 and 2 are expressed as follows:

　

　

 where m^(j) denotes the ship's mass and Iz^(j) the moment of inertia around z'-axis. FVk^(j) is damping forces due to viscous fluid and FEk^(j) the external forces such as propeller thrust rudder forces and so on. Substituting eqs.(21) to (26) into eqs.(27) and (28), the following equations are obtained as: