Brard 



not in principle, but in fact, to non -negligible difficulties even in the field where 

 the equations are linear. 



The first answer given here is therefore faulty. The conclusions of this 

 paper will probably not satisfy fully the naval architects. It is hoped, however, 

 that the ideas developed here may be of some practical interest. 



I. THE FORCES EXERTED ON A SUBMERGED BODY 

 MOVING IN AN INFINITE FLUID 



1. Notations 



Let o'(x',y',z') be a dextrorsum set of fixed axis. The z'-axis is vertical 

 and positive downwards. 



We consider also a dextrorsum set of axis 0(x, y, z) attached to the sub- 

 merged body. 



When the body is in a normal position (that is, when the heel and trim are 

 null), the z axis is vertical and positive downwards. 



The X axis is going from the stern to the bow. is in the middle trans- 

 verse section. Generally, the body is symmetrical with respect to the (z,x) 

 plane. 



The coordinates of referred to the fixed axis are ^,rj, i. 



In order to define the position of the body we introduce firstly a set of axis 

 0(x^,yj,Zj) having its origin at 0, but with the axis 0xj,0yj,0zj parallel to the 

 axis 0'x',0'y',o'z' respectively. We consider three non-Eulerian angles 4j,6,4> 

 (Fig. 1). 



is the head angle. By a (//-rotation about the z^-axis, the x^-axis comes 

 in the (z,x) plane on an axis Ox^; by this rotation, theyj-axis comes on an axis 

 Oy^ . The z 2- axis coincides with the Zj-axis. 



is the trim angle. By a ^-rotation around the yj-axis, the Xj-axis comes 

 on the x3-axis; by the same rotation, the yj-axis and the z2-axis come, respec- 

 tively on axis Oy3 and OZ3. The x3-axis coincides with the x-axis. 



cp is the heel angle. By a (/>-rotation about the x-axis, the y3-axis and the 

 z 3 -axis come, respectively, on the y-axis and on the z-axis. 



The absolute velocity of is V^, of components ^, t], i on the fixed axis. 

 The components of Vg on the x,y, z-axis are respectively u,v,w. 



i is the heaving velocity; the derivatives 41,6,4) are, respectively, the head- 

 ing velocity, the pitching velocity and the rolling velocity. 



818 



