PROFESSOR W. M. HICKS OK VORTEX MOTION. 
71 
bj (say) y (P at^ 9 ), where JX' and JX are both positive. J — is positive between 
0 and X — X'/X and negative between x — X'/X and 1. In the inner spherical nucleus 
(r = 0 to 7' = X'a/X) the stream lines lie between the vortex lines and the meridians 
(similar to the first category). At the interface the stream-lines coincide with the 
meridian. In the outer layer the stream lines and vortex lines lie on opposite sides 
of the meridian for points whose distance from the centre are less than ^ a, or greater 
X® . . X'l x<^^ 
than — For points at a distance - a and a, the vortex lines coincide with the 
X X 
X 
meridians, and between them the two lines lie on the same side of the meridian. In 
the same way the behaviour for aggregates whose parameter is greater than may 
be determined. The periodic nature of the aggregate is again very clearly seen. 
It is perhaps easier to describe the nature of the changes above indicated by supposing 
our eyes placed in a prolongation of the polar axis. Call the vortex lines blue lines and 
the stream lines red lines, and suppose for X small that the stream or red lines lie on the 
right of the meridians. For X = 0, or Hill’s vortex, the red lines lie along meridians 
and the blue lines perpendicular to these, along parallels of latitude. As X increases 
the red and blue lines swing round towards each other, the reds to the right and the 
blues to the left, and this goes on with increasing values of X up to when they 
coincide. Beyond X = X(^’ and up to X = yp the red and blue lines interchange their 
relative positions. In any given aggregate the blue lines move more and more 
towards meridians as we pass from the centre outwards. At a distance — a from 
the centre the blue lines all coincide with the meridians, both red and blue lines are 
swinging round to the left. Beyond the distance a the blue lines cross to the left 
of the meridians and the red lines close up towards the meridians until at the surface 
of the aggregate they coincide with them. 
Between X^* and yp we have doublets. The aggregates lying between XT and XP 
and between XP and XT are however essentially different. 
In the first set in the central nucleus the blue lines lie to the left of the red, and 
both to the right of the meridians for points near the centre. As we pass outwards 
from the centre they swing round to the left, the blue lines swing past the meridians 
whilst at the surface of the nucleus the red lines just reach it. Beyond, in the 
outer layer as we pass out, the blue and red swing further to the left, and later at 
least the red swing back again towards the meridian, coinciding with it at the 
surface. When X = XT red and blue coincide everywliere. They lie to the right in 
the inner nucleus and to the left in the outer layer. 
Between XT and XT we get aggregates in which red and blue lines again change 
sides. In the inner nucleus both lie to the right of the meridian, blue furthest out. 
They close up to the meridian as we pass out from the centre to the nucleus surface. 
In the outer layer the red lines swing further to the left and back again, the blue 
