Brard 



A motion parallel to the horizontal plane is generally impossible when the 

 diving planes are at a zero angle. For instance, the steady motions are, in this 

 case, helicoidal motions around a vertical axis. 



Let us assume that u,w,q are null. 



The arc d' along which the free vortices are shed depends upon the form of 

 the hull. 



1') We consider firstly the case when there are, in the (z,x)-plane, no 

 singularities, appendages and so on, which constrain this arc to be located in 

 this plane. 



a) p =0. Because of the symmetry of the hull with respect to the 

 (z,x) -plane, the arc fl' is in the (z,x)-plane, or in a plane parallel to it. Con- 

 sequently, when the motion is quasi-rectilinear and parallel to the x-axis, the 

 wake surface is approximately parallel to the (z,x)-plane. The previous rea- 

 sonings for the motions parallel to the (z,x) -plane hold in the present case, 

 provided v/u and Lr/U are respectively substituted for w/u and Lq/u. 



b) p + . It is possible that an U-shaped free vortex shed during the 

 small interval (t' , t'+ dr' ) have, at t ' , an orientation with respect to the axis 

 attached to the body different from the orientation at r' + dr' . In this case the 

 summation at t ' of the effects of the free vortices shed during the intervals 

 (t' , r' + dr') cannot be carried out on the same manner as in the case of a mo- 

 tion parallel to the (z,x) -plane. This case occurs, for instance, for a body of 

 revolution with respect to the x-axis. 



2') Let us assume that there are in the (z,x)-plane singularities so that 

 the arc &' is in this plane. 



a) p = 0. This case is quite similar to this of par. 1*) a). It is the 

 simplest from the point of view considered in this paper. 



b) p +0. In this case, if I0I is relatively great, the wake surface is 

 not a plane; it is more or less helicoidal. The nuclei found in the integrals 

 which yield the effect of the wake depend not only upon t ' - t', but also on r', 

 and the expression of the hydrodynamic forces due to the wake is much more 

 complicated than in the case when the motion is parallel to the (z,x) -plane. 



3*) It occurs very often that the singularities mentioned above (par. 2*) 

 exist only on the upperside of the body, and not on the lower side, or inversely. 

 In this case, when p = 0, the wake surface is inclined with respect to the (z, x)- 

 plane. Consequently, the velocities induced by the wake on the body itself and 

 on the rudder and planes generate necessarily forces which components on the 

 z-axis are not null (see Fig. 8). 



If, for instance, Lr/U = 0, v/U >0, and if, moreover, the upper arc of S' is 

 located in the (z.x) -plane, because of the singularities of the hull, but not its 

 lower arc, this latter is located on the portside of the hull; the wake surface 

 induces on the body and on the diving planes located on the stern velocities 



868 



