98 
Proceedings of the Koyal Society of Edinburgh. [Sess. 
an infinite distance, Von Karman suggests that the solution of the problem 
lies in determining the stable arrangement of the rectilinear vortices set 
up behind the body in rows extending off to infinity. 
For our purpose it will be sufficient to admit the motion of a system 
of vortices as stable, if, when they are slightly displaced from their positions 
of steady motion, the displacements do not increase indefinitely with the 
time. If the original displacements be preserved, then the arrangement has 
neutral stability.^' As a first condition the motion of the vortices must be 
steady, and if there is no arrangement in which the vortices set up behind 
the body can move forward steadily, then there can be no possible steady 
motion of the body. For example, by simple considerations it can easily 
Fig. 3. 
be seen that no odd number of rows of an infinite number of vortices 
can move forward parallel to their length so as always to retain the same 
geometrical arrangement. There cannot, therefore, be any steady motion 
of a body having such a shape as to throw off an odd number of rows 
of vortices. 
In the same way there is no steady motion of four infinite rows of 
vortices, so that, on Von Karman’s theory, there could not be any steady 
motion of two cylinders moving along parallel lines, or of a cylinder 
moving parallel to a plane (see fig. 3), the two cylinders being sucked 
together in the former case, and in the latter, the cylinder being sucked 
into the plane. 
Cylinder in an Infinite Fluid. 
The simplest but at the same time the most searching problem of this 
nature is that of the steady rectilinear motion of a cylinder in an infinite 
* For a more rigorous definition of stability, see one given by Prof. Love in Proc. 
London Math. Soc., xxxiii, p. 325 (1901). 
