A Vortex Theory for the Maneuvering Ship 



Figure 8 



which components on the z-axis are >0. Consequently, the variation of Z and of 

 )1I are both >0. This effect is independent of the sign of v U. It leads to a per- 

 turbation of the motion in the vertical plane even when the angle of heel is null. 



4*) When the six parameters u U, w/u, Lq/U, v/U, Lr/U, Lp/U depend upon 

 the time, is it possible to add the effect of the wake due to the three first of 

 them and the effects of the wake due to the three others? The velocities due to 

 one of the wakes may act on the configuration of the other wake. Nevertheless, 

 when the dissymmetry of the body with respect to the (x, y) -plane is not too 

 strong, and, when, moreover, we are in the case of par. 2*) with moderate 

 angles of heel, the velocities due to the wake parallel to the (x,y) -plane are 

 nearly parallel to the wake due to the variations of v/U and Lr/u and inversely. 

 So it is possible, in a first approximation, to obtain the set of hydrodynamic 

 forces exerted on the body in a quasi-rectilinear motion parallel to the x-axis 

 by adding the sets of forces separately found for motions parallel to the ( z,x)- 

 plane and for motions parallel to the (x,y) -plane. 



In paragraphs 10, 11, 12, we restrict our analysis, for reason of simplicity, 

 to the cases when such an addition is allowed (however, in par. 12.4, we will 

 consider a more general case). 



We set 



/t'), (^)^^ . f„,(t'), (^j^^ = f,,(t') 



Lq 



^vL, 



f.3(«'>. (^),,= fo.ff). !(^),, = !f„(f)- 



and assume, in par. 10, 11, 12, that these functions have negligible squares and 

 products. 



869 



