101 
1914-15.] Resistance to Motion of a Body in a Fluid. 
with the time, and therefore, according to our criterion for stability, such 
an arrangement of vortices is unstable. 
Following immediately upon the displacement of g = 0, of course, the 
remaining vortices in the two rows proceed to displace themselves also 
from their positions of steady motion, and react slightly differently upon 
the motion of the vortex q = 0. But these disturbances and consequent 
reactions are evidently all of the second order at least, and therefore need 
not be considered in comparison with first order quantities. 
As a mere proof of instability, the displacement of a single vortex from 
its steady position is evidently a sufficient test, but in the case of a stable 
arrangement the disturbance would require to be perfectly general. 
If there is a stable distribution of two infinite rows of vortices — and it 
is not a priori evident that there must be — it must be the arrangement (5), 
fig. 4. Von Karman asserts that this is so. 
Taking the origin of co-ordinates (see fig. 5) midway between the 
vortices p = 0 and q — 0 when in the position of steady motion, suppose 
the vortices are each given small displacements to the q^^^ vortex on 
the one row, and {^pr]p) to the on the other row. 
The co-ordinates of the vortices at time t = 0 will then be given by 
^q~ 9.^ — + pi 
h h 
2 +"^^ lip 
We proceed to consider the motion of the vortices ^ = 0, p = 0, whose 
co-ordinates we call 
' I 
, h 
Vo - 2 . 
r I p 
4 + = 
h , 
and 
