PROFESSOE. W. M. HICKS ON VORTEX MOTION. 
G5 
Before discussing tlie value of v it will bo well to get some general idea of the 
nature of the inotions. One of the most striking peculiarities of these aggregates 
is the quasi-periodicity of type as X increases from 0 to inlinity. The best way to 
illustrate this is to use a graphical construction. Now 
i//oc {J (Xx) — a;" J (X)]. 
In fig. 2, Plate 1, the curve y = J (X) is drawn. Pj corresponds to a given type 
(X) of aggregate. A parabola is drawn with vertex at O and passing through Pj. 
Bepresent any abscissa to the left of X (or of Pi) by Xx, where a;<l. Then the 
differences of ordinates between the curve and the parabola up to P represent 
J (Xa;) — a;“ J (X). 
It is clear from the figure that, in the position Pi, this function never vanishes for 
x<\. In the second position, P 2 , however, the parabola intersects the curve at 
another point p. For this point (suppose x = Xq) ijj vanishes for all values of 6, and 
the corresponding current sheet is a sphere internal to the boundary. The aggregate 
consists of two portions with independent motions. The primary circulations are in 
opposite directions, and there will be tivo equatorial axes. So, as P moves on 
along the curve, i.e., as X increases, we get families of aggregates with three, four, &c., 
layers, and a corresponding number of equatorial axes. We shall denote any transition 
value of X by Xo. Each layer will have its own secondary circulation, given by the 
circulation round the double circuit formed by its equatorial axis, and an equator on 
its boundary. 
Now the secondary spin velocity is given by 
vp sin (b = 'A* 
' ZTra 
And since = 0 on the boundary, it follows that 
X, 
V,, = 2 wpv sin (f), along the equatorial axis only, = — 
cc 
where is the value of if/ at the nth equatorial axis, or 
irp 
Six — sin X 
— xlJ'}, 
where J„ = J (Xx„) and J' = J (X). 
VOL. GXCII.—A. 
K 
