A Vortex Theory for the Maneuvering Ship 

 in-phase forces and moment, we get 



2W 

 AL 



— Cfl+fl'^), IJ.[^, ^;5+M^G' ^Is-^^^G- ^^2+^-><'i 



In many cases, these five coefficients will be either nearly constant or small, 

 and the calculation of the g -functions will be perhaps without practical interest. 



Similar reasoning based on the results of the harmonic tests with planes 

 would show the possibility to obtain rp. 



Practically all the unknown coefficients and functions may be determined, 

 except those which are connected with the variations of u/u. In its present 

 state, our Planar Motion Mechanism is unable to yield them, because no sinus- 

 oidal motion parallel to the x-axis is possible. But it is to see that the system 

 could be modified for that purpose, if necessary. 



19.2. Motions in the (x,y) -Plane 



From pars. 10-12, we could deduce, for this family of motions, formulae 

 similar to those of par. 8, and we could show, in the same manner as in 19.1, 

 that harmonic forced motions in the (x,y) -plane give also the numerical values 

 of the coefficients and functions which are needed to write the equations of the 

 motion in the (x,y) -plane, or more generally, of any motion, provided the ex- 

 pressions of the forces are additive. 



20. Effects of the Non-Linearity and Other 

 Sources of Errors 



20.1. Non-Linearity 



The so-called "true" equations are true only in the linear field. The non- 

 linearity may affect many points of the semi-theoretical views explained in this 

 paper. Some of them are related to the part of the quasi-steady motions theory 

 which we use in our formulae. Some others concern specifically the structure 

 of the wake and the method used for taking its effects into account. 



1*) Because submerged bodies are generally very poor lifting surfaces, 

 the coefficients a, b, a' , b' , for the motions in the (z,x) -planes, a^, bj, aj, 

 bj, for the motions in the (x,y) -plane, are not really constant. A question would 

 be to know whether it is possible to substitute for their expressions versus the 

 drift angles e or S such expression as a.°s for a8 + aS|s| or for ah + aS^. 



We have also to observe that our integrals 



0(t '- t') dr', . .. , 



I 



become 



901 



