A Vortex Theory for the Maneuvering Ship 



5.3. Another Expression of the Set of Forces Due to 

 the Velocity Potential 'i' 



The absolute momentum due to the distribution of normal doublets 



J_ 



4^ d^(M) ^ 



on a small surface d:^^^ is 



it acts through point /i. 



Consequently, the total set of hydrodynamic forces exerted on the body may 

 be obtained by starting from the variations during the interval ( t ' , t ' + dt ' ) of 

 the absolute momentum due to the distributions of normal doublets on S and on 

 the wake surface. These momentums are additive. Consequently, the set of 

 forces (?j) comes from the velocity potential: 



2 

 k=0 



while the set of forces (J') comes from the velocity potential: 



2 



k=0 



Here, we deal only with the set of forces (?'). The set (J.) will be ob- 

 tained in par. 5.4 by another way using the absolute kinetic energy due to $. 



Let us consider the case "bj." 



At time t ' the absolute momentum of the fluid is 



Qi(t') = Q;(t') + o';(t') . 



OJf t ') is due to the doublets distributed on the wake and Oi'( t ') to those dis- 

 tributed on s. 



The general resultant of the hydrodynamic forces due to w ^ is 

 One has 



851 



