PROFESSOR W. M. HICKS ON VORTEX MOTION. 
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certain point Q where the parabola touches the curve. It then changes sign and 
remains negative until P reaches another Q point where the parabola again touches 
the curve, and so on. 
We shall call the values of X corresponding to the Q points the X 2 values, and 
denote them in order by X^P, . . . Xp\ Thus for values of X < XP the aggregate 
moves in the direction of the rotational flow up the axis. At X = XP the aggregate 
is at rest, the velocity of the fluid on the boundary is zero; as X increases beyond 
this, the aggregate takes on another layer with primary rotation in the opposite 
direction, and it moves in the fluid in a direction opposed to the rotational motion of 
the innermost layer. It regredes relatively to this. The velocity at first increases 
and then diminishes until P reaches the second X., point, when the corresponding 
aggregate is at rest in the fluid, and so on. 
The periodic nature of the aggregates is thus evident. We get for example a 
whole periodic family of aggregates whose peculiar property is that they remain at 
rest in the fluid. The members of the family differ, amongst other things, in the 
number of independent layers each possesses. 
So we get another family formed by values of X, corresponding to points where the 
J-curve cuts the axis of x. We will call values of X, corresponding to these the X^ 
parameters, and denote the orders in the same way as for the Xo parameters. As we 
shall see shortly, the distinguishing property of this family is that in each of them 
the vortex lines and the stream lines coincide. 
For small values of X it is preferable to express the value of U in terras of the 
lowest powers of X. 
It is easy to show that 
StX — sin X = -d;. X' 
■7 
6,0 
-i-r.- 
2n 
Xin+i 
2 ft + 1 (2 ft + 1)1 
whence 
u= c (I -Thn 
5a 
This gives for X = 0 the value of U already known for Hill’s vortex. 
The curve y = U/Uq, where Uq is the velocity of the non-gyrostatic aggregate of 
same cyclic constant and volume, is drawn in fig. 3, Plate 1, up to XP. The periodic 
quality is evident. 
21. The directions of the lines of flow and of the vortex lines are given by 
, vp sm (b 
tan (b = ' 
vp cos 
tan X — 
sin X 
wp cos X 
A, yjr 
a 
dn 
X 
a 
dn 
J shd 0 
(J — x-J') sill- 6 
}■ 
(24), 
