PROFESSOR W. M. HICKS OK VORTEX MOTTOK. 
35 
distinguishing feature common to all the members is that the stream lines and 
the vortex lines are coincident. 
The parameter k defines the total angular pitch of the stream lines, on the outer 
current-sheet, viz., up the polar axis and down the outside ; although in the aggregates 
with more than one axis these lines are not one continuous stream line. The first 
aggregates—with X< 5 ‘ 7 r )37 (the first X2 parameter)—behave abnormally. Beyond 
these we get successive series, in one set of which the velocity of translation is in the 
same direction as the polar motion of the central nucleus, in the alternate set the 
velocity is opposite, and the aggregate regredes in the fluid as compared with its 
central aggregate (see fig. 3, Plate 1). The physical analogue of these aggregates 
is obvious. It is specially enlarged upon in the abstract. '" 
Suppose we set ourselves the problem of making a set of aggregates with greater 
and greater angular pitch. As we do so we shall find that as the pitch increases 
the equatorial axis contracts, and the surface velocity diminishes. On the outer 
layei’s (ring shaped) the spiral is chiefly produced on the inner side facing the polar 
axis, until on the boundary itself the stream lines flow in meridians, and the 
twist is altogether on the polar axis. The pitch can be increased up to a certain 
degree. As this is done, the stream lines and vortex lines fold up towards one 
another, coincide at a certain pitch, and exchange sides. When an external angular 
pitch of about 330 ° is attained it is impossible to go further if a simple aggregate is 
desired. If a higher pitch is desired it is attained by taking it in two parts. First, 
a central spherical nucleus of the same nature as the former, in which a portion of 
the twist is produced, and outside this a spherical shell, in which the spirals have 
the same direction of twist, and complete the pitch to the desired amount but in 
which the spirals are traversed in the opposite direction. With increasing pitch 
this layer becomes thicker, and its equatorial axis contracts relatively to the mid¬ 
point of the shell until another limit is reached ; the stream and vortex lines again 
fold together, cross, and expand as this second limit is reached. If a larger pitch still 
is desired there must be a third layei', and so on. The first coincidence of vortex 
and stream lines takes place for an aggregate whose pitch is 257 °’ 27 '. Whenever a 
maximum pitch is attained the aggregate is at rest in the fluid. This is first 
attained for an external pitch of 330 °' 14 '. Beyond this there are two equatorial 
axes. For an external pitch of 442 °' 37 ' the stream and vortex lines again coincide, 
the internal nucleus gives 257 °' 27 ' of the pitch and the outer shell the remainder, 
and so on. 
At the end a theory of compound aggregates is developed similar to that in 
Sect. ii. for non-gyrostatic vortices. It is not worked out in detail in the present 
communication, but the conditions are determined for dyad compounds, whilst a 
similar tlieory holds for triad and higher ones. Each element of a poly-ad may consist 
of singlets, doublets, &c. The equations of condition leave three quantities arbitrary— 
* ‘ Roy. Soc. Proc.,’ vol. 62, p. 332. 
