99 
1914-15.] Resistance to Motion of a Body in a Fluid. 
fluid. The body in its motion continually throws off vortices along two 
lines trailing behind it, parallel to the direction of motion, and our problem 
is to determine the arrangement giving both steady and stable motion of 
the vortices. I propose to discuss the stability of the vortices at a great 
distance behind the moving body, so that, instead of dealing with two semi- 
infinite rows, we may, following Von Karman, suppose the vortices extend 
to infinity in both directions, and our problem reduces to discussing the 
stability of two infinite rows of vortices. There are evidently only two 
possible arrangements such that the vortices move forward steadily, (a) 
and (b) (see fig. 4). 
(ou) 
f -- 
(-&) 
5 
J7 
P 
■ ^ - 
Fig. 4. 
That (a) is probably an unstable arrangement is indicated a priori from 
the fact that, if we increase the distance between a pair, all the vortices 
behind will at once tend to shrink and shoot through those in front. A 
simple proof may, however, easily be given as follows : — 
Suppose the origin of co-ordinates be so chosen that the vortices are 
situated at the points {^l , and where ^ and q are all integers 
between -1- oo and — oo , and suppose the vortex g = 0 is displaced to the 
point ^^ 0 , ^ -f 71^ where and are small. If -f f and — f be the 
strengths of the vortices in the first and second rows respectively, the 
component velocities of the displaced vortex are given by 
