Brard 



Z^Ct') = - ipAU^a, |(^)^^ 0(0) + j (^) ^ _i_0(t'-r')dr'| , 



y (41) 



and so on. 



In par. 8, the notations used here will be slightly modified. We will use a, a' 

 instead of aj,a{ and b,b' instead of a2,a2. 



5.4. Set of Hydrodynamic Forces Due to p. and p'^^) 



When the body is symmetrical with respect to the (z,x) -plane, the kinetic 

 energy of the fluid in the absolute motion due to the potential 



^00+^0 u 



IS 



Lq 



+ *01 [fj}^,^%2 llfj,,+'^03 (U)., + ^04 (if) , + ^05 VU 



Lp 



t ' 



2T = pW{Aii(U+u)2 + yLijV^ + fx^v/^ + 2Mi3(U+u)w 



- 2L [vjjwq + Vj5(U+ u)q + 1^34^? + '^26^'"] 

 + L2 [\jp2 + X^qS + x.^r^ - 2\i3prJ} , 



where W is the volume of the body. 



The components of the set of forces X., Y^, z. , £. , !)il. , }l. are given by 

 the Lagrangian expressions: 



Writing before the accelerations /Xj , ... for /j,^, . . . , in order to take into 

 account the forces due to the pressures p'^^\ and assuming that u/u, w/u, 

 Lq u are of negligible squares and products, we get, in the case of a motion 

 parallel to the (z,x) -plane: (with 



z; = E < 



(i) 



and so on): 



860 



