Brard 



Let o'zj.o'xj be fixed axis respectively directed as the z-axis and the 

 X-axis. 



Let bj.bi be the coordinates of in the new set of axis. We have 

 In the purely heaving motions here considered, one has: 



0(6?2) ; 



this product is therefore negligible. Moreover 



w 1 d^ 



u - ^^ Ud? 

 is a sinusoidal function of the time. Therefore, we set: 



i[ = 6^(0) + Ur't + z + Zq cos cot (z - constant, z^ >0, 



and obtain 



constant 



w — ■^O coh I n 



Lw 



U2 



'0 /a)L 



L \ U 

 In these motions, the components of the set of forces are 



cos cot . 



(4) 



Z = Z + Zj^ cos OJt + Zj^ cos [OJt + -p: 



X = X 



+ Xj^ COS ojt + Xj^ cos [ajt + ^j 



))I = I + \ cos a)t + \^^ 



cos \wt + — 



(5) 



The subscript 1 is relative to the components in phase with the motion; the 

 subscript 2 is relative to those which are out of phase with the motion. 



C. Purely pitching motions 

 In these motions e is a sinusoidal function of the time: e 



+ 6q cos a)t, 



( = constant, t- ^ >0); moreover, O'zJ, O'xj being fixed axis, with Cz',Zj) = 6; 

 the ordinate l\ = ^c^ + ^ is also a sinusoidal function of the time chosen in such 

 a way that the angle of drift be a constant. Consequently, the purely pitching 

 motions here considered are defined by the conditions 



884 



