36 
PROFESSOR W. M. HICKS ON VORTEX MOTION. 
as, for Instance, ratio of volumes, ratio of primary cyclic constants, ratio of secondary 
cyclic constants. The full development of this theory is. however, left for a future 
communication. It is clear that spiral or gyrostatic vortex aggregates are not confined 
to forms symmetrical about an axis. Their theory is however much more complicated. 
If we take any particular spherical aggregate with given X and primary cyclic 
constant (p.), the energy is determinate. We may, however, alter the energy. If it 
be increased, the spherical form begins to open out into a ring form, whose shape and 
properties have not yet been investigated. If the energy be increased sufficiently 
the aperture becomes large compared with the thickness of the rotational core, and 
approximate calculation can be applied. The differential equation for ijj is given in 
Sect, i., but its development is left for a future occasion. After that I hope to deal 
with the question of stability, and then more fully with that of the conditions of 
combination. The new field opens up so many questions of interest that other 
workers in it are welcomed. 
Section i.— General Theorems. 
1. To give an idea of the nature of the rirotions considered in the present investi¬ 
gation, consider the case of motion of an infinitely long cylindrical vortex of sectional 
radius a. The velocity perpendicular to the axis inside the vortex will be of the 
form V — f{r), where /{O) = 0. Outside it will be given by v = Yaj‘}-, where Y =_/(«). 
We may, how^ever, have a motion in wdiich the fluid moves parallel to the axis 
inside the cylinder with rest outside. The velocity will be of the form n = F (r) 
inside, wdiere F («) = 0, and zero outside. Both/"(r) and F (?•) are arbitrary functions 
subject only to the conditions / (O) = 0 and F {«) = 0. 
Putting aside for the present the question of the stability of these simple motions 
or of their resultant, it is clear that if we supei'pose the two we get another state 
of motion in which we have vortex-filaments in the shape of helices lying on 
concentric cylindric surfaces. The problem to be considered is whether it is possible 
to conceive a similar superposition of two motions in the case of any vortex aggregate 
whose motions are symmetric about an axis. 
There are an infinite number of either ring-shaped vortices, or singly connected 
aggregates (of which Hill’s vorte.x may serve as a type), differing from one another 
in the law of vorticity of the ^different parts—the most important being those in 
wdfich the voi’ticiiy is uniform. The motions in all these are known in terms of 
the stream function i//. The value of i// is however at present only actually known for 
an infinitely thin ring-filament or for a spherical aggregate. 
2. We are to consider two superposed motions. The one component is in meridian 
planes through an axis and can be defined in terms of the stream-function The 
* Throughout is taken as the total flow up througfli the circle whose radius is />. In other words tlie 
velocity perpendicular to ds is —• 
