PROFESSOR W. M. HICKS OH THE THEORY OF VORTEX RINGS. 729 
r' = inner sectional radius 
x = (r'/r) 3 
r 0 = value of r when the aperture is infinite 
a = radius of ring 
a 1 = radius of ring when the hollow begins to form 
r x = sectional radius ,, ,, „ „ 
V = velocity of translation 
Section I.— Preliminary. 
1. The motion we are about to consider is one of steady motion in a fluid, part of 
winch is rotational. If xp denote the stream function, co the angular rotation, and p, z, 
the cylindrical co-ordinates of a point, then 
b 2 x}r .b~yjr 1 bifr 
~b^ + I^~p'bp = 201p 
Now when the motion is steady the rotations are so arranged that the vorticily is a 
function of the stream line or 2 oj/ p=f(\jj). For steady motion then we have 
b~yfr b~\y 
~b# + ~b^ 
1 b'\Jr 
P h P 
=pV( x I j ) 
Before, therefore, it is possible to discuss the properties of any vortex ring it is 
necessary to know its vorticity. The case considered in the following pages is that 
where the vorticity is constant. The methods developed will however apply to any 
cases where the motion is arranged in anchor-ring shells, the vorticity in any shell 
being constant. Here then f(\p) is a constant = A (say). A particular integral is then 
at once obtained, viz., i//=-|Ap 4 , and the general solution becomes [I. § 1]. 
*/>—-g-Ap 4 + So (A b R»-J- BAh) cos (nv- |-a„) 
Since the translatory motion is uniform the problem may be reduced to one of 
steady motion by impressing on every point a velocity equal and opposite to V, the 
velocity of translation. The stream function for the fluid outside the core will then 
be of the form 
^ 2 = — -^ X o A/R,, cos nv .(1) 
whilst for the portion of fluid constituting the core it is of the form 
xp i =lA P ^+-y=^ o (A n U n +B u T n ) cos 
nv 
( 2 ) 
