Brard 



Moreover, the components p,q, r of the angular velocity of the body on the 

 moving axis are 



p = (p- \p sin 0, q = cos (p + ^ cos 6 sin cp, r = \p cos 6 cos (p- 9 sin cp • (2) 



Let G be the center of gravity of the body. Let Sm be the mass of a small 

 volume sw which coordinates with respect to axis parallel to the x,y,z-axis, but 

 having their origin at G, are x",y",z". When the body is symmetrical with re- 

 spect to the (z,x) -plane, the moments of inertia of the body are 



Ij = 5:(y"2+ z"2)Sm , I ^ ^ ^(z" ^ + x"^)hm , I3 = ^(x"^ + y"2)Sm , 



(3) 



I J2 - 2( z"x")Sm . 



Let p be the specific mass of the fluid, and m the mean density of the body 

 with respect to the fluid. We introduce dimensionless coefficients by the 

 formulae: 



I J = PWlVxj , I2 = pWLVxj - 13= pWlVx3 , Ii3 = /3WlVxi3 , (4) 



where l is the length of the body along the x-axis. 



When G is not at 0, its coordinates are L^q,0,L(,q. We assume that ^q, ^q 

 have negligible squares and products. Otherwise, the moments of inertia of the 

 body about the (x, y, z) -axis would be 



I{ = pWL Kxi + ^G^) , I2 = pWL^/LtXj , 



(5) 



i; = pWLV(x3 + ic) , Ija = pWlV(Xj3 + Cg^g) ■ 



The set of the absolute forces has a general resultant ? and a resultant 

 moment i referred about the origin of the axis attached to the body. One has: 



5 = Xi^ + Yiy + Zi^ , I = £i^ + My + m^. (6) 



In this paper, we don't consider the relative forces, that is the forces in the 

 set of axis attached to the body. 



2. Some Particular Motions 



Motions Parallel to the ( z , x) -Plane — The y -component v + rx - pz of the 

 absolute velocity of any point attached to the body is null. Consequently 



v=0,r = 0,p = 0. (1) 



Therefore 



4> ^ 4j sin e = Lg cP tg 6 , q = . (2) 



cos <p 



820 



