Brard 



10. The Absolute Forces Due to the Velocity Potential 

 (case of par. 9) 



The velocity potential in the absolute motion is 



5 5 



0(M,t') = %,(U) + ^ %^(M) fok(t') + 2] VM.t') 



Potentials o are those which are found alone when the fluid is quite perfect. 

 Potentials ^^ yield the effect of the wake. This formula involves the hypothesis 

 at the end of par. 9. However, in the present paragraph, we don't consider the 

 effects of the wake on the appendages (see par. 11). 



10.1 Effects on the Wake on the Body Itself 



The contribution of f^^ct') in Vj.(M) is 



VsE(M,t') = -iyPz(M) + i^py(M) . 



(Vgn)jjj is generally very small when m is on S. For this reason, we neglect 

 here the contribution of fgj. 



As seen in par. 5.3, the wake has an effect on the apparent forces of inertia. 

 This effect will be taken into account in par. 10.2. 



For reasons of simplicity, we will change slightly the notations of par. 5.3. 

 We use here the following symbols: a for the lift due to w/u ; b for the lift due 

 to Lq/u; a' for the moment due to w/U; b' for the moment due to Lq/U; (aj,aj) 

 and (bj,bj), are substituted respectively for (a, a') and (b,b') in the terms coming 

 either from v/u or from Lr/u. 



Moreover, we assume, as in par. 5, that a set of three functions 0, 0, (p' is 

 sufficient for yielding all the effects of the delayed circulation when f^g, fg^ are 

 null; and in the same way that a similar set 4)^,4j^,4^[ gives this effect when 

 foo'foi'fo2 are null. 



Lastly, we admit that the £ -component of the moment, when fga- ^04 are 

 different from zero, may be expressed by means of two coefficients m,m' . 



Finally, that leads, for quasi-steady motions, to the set of forces 



870 



