1096 
the current originally closes, and that it moves further in the 
direction of the axis of the current behind the body. Approximately 
it will be confined to a small paraboloid round that axis and the 
whole vortex region may be roughly represented by fig. 15. 
In the limiting case R=o the vortex layer at the body will 
be infinitely thin and the paraboloid will contract to the axis (see 
fig. 16). The vortices from opposite parts of the surface of the body 
having opposite signs, they will soon vanish in the axis. [nthe 
limiting case R= we have therefore everywhere outside the body 
irrotational motion, while at the surface of the body an infinitesimal 
vortex layer is found’). The potential p of the current is determined 
by the ordinary condition: 
ae siet tn Sie AREA 
On 
for the absolute flow along the whole surface of the body. When 
we wish to obtain a corresponding representation of the vorticity 
distribution and yet to use as in Osnrn’s solution a linear equation, 
equation (4) must be replaced by: 
0 
where v, is written for a known current, which at a great distance 
from the body approaches the parallel current U, following however 
the surface of the body in its immediate neighbourhood. 
For v, we may e.g. take the ordinary irrotational current. In this 
case (20) changes into an equation applied by Boussinesq in the 
calculation of the transport of heat by a moving fluid’). It is 
d00p dal? 
the -front; REN ma (nx) at the back of the body. Behind the body we 
therefore have: 
i zoja n= cos (nat). 
ae noon 
As soon as the body is x into motion we shall have over the whole surface 
0, 
= = U cos (nx), so that ¢ is the ordinary potential; the space outside the region 
n 
described by the body is then the whole space outside the body, hence every- 
where v = Vo. 
1) Perhaps the a-axis behind the body must be regarded as a singular line in 
the flow. 
2) J. Boussinesq, Journa de Liouville (6) 1, p. 285, 1905. See also: A. RUSSELL, 
Phil. Mag. (6) 20, p. 591, 1910, and L. V. Kine, Phil. Trans. London A 214, 
p. 373, 1914. 
