60 
PROFESSOR W. M. HICKS OK VORTEX HOTTOK. 
Impress — V on every point, that is, deduct ira'Yr-Z.y. Then the stream- 
iiinctioii referred to the boundary is 
- 1A {a - rf + I {cj? - r^-) Z,. 
At the outer boundary i/; = 0. If we trace the stream-line i/; = 0, it is seen that 
it consists of the circle r — a and the curve 
iA (« — r) =: Att'-F?’- {a + r) sin" 6. 
This passes through the poles (r = a, 6 — 0) and touches the circle there. Hence 
the space between this and the outer boundary does not contain the polar axis. The 
motion given by r/; is therefore finite and continuous there. The space inside it 
must be excluded as giving a motion not possible—or rather, a motion due to sources 
and sinks on the polar axis. AVe shcdl suppose it excluded by replacing the fluid by 
a solid nucleus of the shape required. 
The radius of an equatorial axis is given by d\p/dr = 0 when 0 = tt, 2, or by 
A {a — r) q- ^TrFr {cd — 2r-) — 0, 
5A 
In this wuite r/a = x and , = h. Then 
lOTT-rff- 
(h — 1) X — h = 0 .(13). 
This has one root between 0 and 1. The other roots must either be both imaginary, 
or, if real, one at least must be negative, since the coefficient of x- is zero. As, 
further, x = — co and x — 0 both make the expression on the left of the same sign, 
both these roots must be negative. Hence there is one and only one root between 
0 and f. That is, there is only one equatorial axis. 
In the special case b = the radius of the equatorial axis is a. 2~^ — •7937u. 
For this curve 
>/'. = iA | q («= - r-} Z, - (a - r)=j. 
The curves are drawn in fig. 1, Plate 2, for values of 2\fjjAa^ =■ — '1, 0, '1. 
The value at the equatorial axis is '397. The value (— T) is drawn to show how the 
discontinuity enters. 
The velocity along a parallel of latitude is given by the equation 
2Trpv sin (ji = f — ^(2 Ai/j). 
This is zero at the surface and on the spindle-shaped nucleus, and increases to a 
maximum at the equatorial axis. The secondary cyclic constant is the circulation 
