Brard 
Similarly, the fluid point P which was at Bo. at t' is at t ata 
point ly 41, and 
" 
= W 
Bo I , ee (Pee) dee (5. 34) 
The fluid points which were at t' onthearc Bo, By ,t! infinitely 
close to B oP 1,t 1 areat t onanarc Jy tly 4. Let ty be the 
time of the beginning on the unsteady motion, e suppose that the 
y-component of ie has on the arc ie j Piss , a constant sign (po- 
sitive) during the time interval [t,, t) (Fig. a aan But it might occur 
that this sign changes at times t, , tz,... We also suppose that this 
sign does not depend on the abscissa X of B,. The most general 
case is still more complicated and will not be examined in this paper. 
The points Ty describe a line starting from IX to and 
ending at Bo. . Similarly the points Jy describe a line starting 
from Jx, to and ending at Bi, re These lines generate a surface 
doe and a surface de, ; u smerd ci The arc Jx, t! yy t1 gene- 
rates a surface Sy ‘beginning at Jy, ie Ty ts and ending on the arc 
Bo By . There Ei lay: So. is one of the edges of Dt and the 
e ayy IE 2? fe) 
arc Ito So. is one of the wm ea8 of oe . The fluid points which 
wereat t! on So, Si. describe a sea when t' varies from 
t= to) os. On oy» , at t' , we had on the port side pate a bound vortex 
fo) 
filament of intensity dy I, (X,t') on the arc BoB, 41, and on the 
starboard side jogs a bound vortex filament of intensity dy J (X, t') 
with the support B, | 4: By . During the interval (t')..ot' + de')etthe 
variations of these two intensities are dj: dyT, (X,t') and 
dy: dy r, (X,t'). It is because they are not equal to each other that a 
free vortex is shed. At t' the support of this free vortex is Bie ! Bo 
e 
and its intensity in the direction from By, to Bo. is equal 
e; t! 
to dir uly (x5 t') = r, Cx; ")] . Because the intensity of a free vortex 
is time-invariant, the intensity of the free vortex filament whose sup- 
portis Jy 4: ly 4, is at t also equal to dy: dy i ea r2(x, | : 
The closed contour Bo, aba Jy t! Ty t! Bog is thus the support 
of a free vortex filament of intensity dj: dy E (33,/¢") -T, (x; ) ‘ 
1226 
