34 
PROFESSOR W. M. HICKS OK VORTEX MOTION. 
The chief part of the following investigation (Sects, i. and iii.) was undertaken with 
tlie view of discovering whether it was possible to imagine a kind of vortex motion 
which would impress a gyrostatic quality which the forms of vortex aggregates 
hitherto known do not possess. The other part (Sect, ii.) deals with the non- 
gyrostatic vortex aggregates, the discovery of which we owe to Hill,* and investi¬ 
gates the conditions under which two or more aggregates may be combined into one. 
It is shown that it is allowable to suppose one or more concentric shells of vortex 
aggregates to be applied over a central spherical nucleus, subject to one relation 
between the radii and the vorticities. In all cases the vorticities must be in opjiosite 
directions in alternate shells. The special case when the aggregates are built up 
of the same vortical matter is considered, and the magnitudes of the radii and 
the positions of tlie equatorial axes determined. The cases of motion in a I’igid 
spheroidal shell and of dyad spheroidal aggregates are also considered. 
The chief part of the paper refers to gyrostatic aggregates. The investigation 
has brought to light an entirely ne\v system of spiral vortices. The general con¬ 
ditions for the existence of such systems, wdien the motion is symmetrical about 
an axis, are determined in Sect, i., and are worked out in more detail for a particular 
case of spherical aggregate in Sect. iii. It is found that the motion in meridian 
planes is determined from a certain function xfj in the usual manner. The velocity 
along a parallel of latitude is given by v —/[xjj] / p where p is the distance of the 
point 1‘rom the axis. The function rjj, however, does not depend on the ditferential 
equation of the ordinary non-spiral type, but is a solution of the equation 
1 d-yjr Cot 6 d-y^r d/’ 
7 /;.r + v' pffz — = p ^ 
where F and f are both functions of i//. The case F and fdfjd^li both uniform is 
briefly treated. It refers to a spiral aggregate with a central solid nucleus, and 
is not of great interest. Tlie case F uniform and fee \jj is treated more fully. If 
^'= \\fj/a where a is the radius of the aggregate 
The most striking and remarkable fact brought out is that witli increasing para¬ 
meter X. we get a periodic system of families of aggregates. The members of each 
family differ from one another in the number of layers and equatorial axes they 
possess. I have ventured to call them singlets, doublets, triplets, &c., in contra¬ 
distinction to the more or less Fortuitous and arbitrary compounds dealt with later, 
and which I have named monads, dyads, triads, &c. Of these families two are 
investigated more in detail than the others. In one family (the X.^ family) all the 
members remain at rest in the surrounding fluid. In the other (the Xi family) the 
// = A J., 
JoX [ siir^. 
Uii ti Sjiberical Voi tc.v,” ‘ Phil. Tnnit;.,' A, vol. 185, 1894. 
