107 
1914-15.] Resistance to Motion of a Body in a Fluid. 
Using the relation ^ = cosh~^ ^3 in (30) and (31), A and A' become 
V/2 
-oo-^ 
+ GO p 
a=2:A 
and therefore from (63) 
A' = 0 
+ 00 , p 
(64) 
and the motion is seen to be unstable for all disturbances given by P 
except those for which 
+ CO p 
-CO 
for example when P is an odd function of p, in which case the equilibrium 
is neutral. 
By subjecting the system to a particularly simple disturbance we have 
found that j = cosh ^3 is a necessary condition for stability, but that 
even when this condition is satisfied the arrangement does not remain 
stable to more general types of disturbances. 
Contrary, therefore, to Von Karman’s conclusions, there is no com- 
pletely stable arrangement of the vortices created behind the moving body. 
They persist in the arrangement (h) for some time after creation and for 
a very short distance behind the moving body, but they finally break up 
and diffuse through the fluid as vorticity generally. This is no doubt the 
real explanation of the photographs taken by Von Karman and Rubach,^ 
showing the formation of half a dozen vortices in each row in arrangement 
(b) behind the moving body. The problem is evidently much more compli- 
cated than Von Karman’s theory suggests. It would seem that for a short 
distance behind the moving body a free surface is momentarily set up, but 
that in consequence of its instability it immediately breaks up and gives rise 
to the series of vortices. For a short time these persist, but finally they also 
break up as indicated, and the difficulties of the problem only commence. 
I desire to express my gratitude to Professor A. E. H. Love for his 
helpful and instructive criticism during the progress of this investigation. 
Appendix. 
In the foreofoing discussion the sums of certain Series which occurred 
o o 
were assumed to be known. These series may be simply and quickly 
summed in the following manner. 
* Phys. Zeitsch., Bd. xiii, 1912. 
