Brard 
If S >0, this equation has one and only one,root. If S< 0, there 
exist three roots for small values of = phen mt and, in particu- 
lax, for ,@® v= 0... Seer cae ima\sne hee ate == ), be the 
three roots with (+=), 4 Ss ae J, . It is to be expected 
2R 
that the steady motion 
sa )o is unstable. 
This is confirmed by the study of the equations of the un- 
steady motion when Y, and N, are neglected. After a perturba- 
tion, the stable steady motion is reached again without oscillation. 
When one takes into account Y> and N, , that is the terms 
depending on the history of the motion, one sees that, if S > 0, the 
steady motion is still stable, but, after a perturbation, it may occur 
that the transient motion be oscillating. It may even occur that no 
straight motion be possible for © equal to zero ; the head is cons- 
tant in the mean, but it is continuously oscillating. 
Oscillating motions in calm water are therefore a conse- 
quence of the delayed circulation around the ship. They appear when 
the ship has to proceed a long path before the circulation becomes 
close to its asymptotical value. 
In spite of the rather rough assumptions involved in Casal's 
theory, it appears that this theory is qualitatively in good agreement 
with experiments, except for what concerns the position of the result- 
ant force in the oblique translations. According to the above expres- 
sions for Y and N when R= , this force should intersect the 
plane of symmetry at a point practically invariable and located inside 
the ship. Experiments on models show, on the contrary, that this 
point can be located ahead of the bow for very small angles of attack. 
Then,when the angle of attack increases, the & of this point rapidly 
decreases and, finally, takes a value rather close to that assigned by 
the theory. 
The explanation of that discrepancy seems to be that the 
free vortices are shed along the stern-post and not along the keel line 
when a is very small. Because self-sway motions are very undesi- 
rable, attention is to be paid to this point. That is also for this rea- 
son that we have indicated above the existence near the bow of a very 
strong vortex represented, ina first approximation, by a 6 -function. 
In the past, the wanted maneuvering qualities mainly con- 
cerned the characteristics of the motions at large rudder angles. 
1258 
