62 
Proceedings of the Iioyal Society 
vortex motion with rotational moment. To illustrate the second 
steady mode, commence with a circular ring of flexible wire, and 
pull it out at three points, 120° from one another, so as to make it 
into as it were an equilateral triangle with rounded corners. G-ive 
now a right-handed twist, round the radius to each corner, to the 
plane of the curve at and near the corner; and, keeping the cha- 
racter of the twist thus given to the wire, bend it into a certain 
determinate shape proper for the data of the vortex motion. This 
is the shape of the vortex core in the second steady mode of single 
and simple toroidal vortex motion with rotational moment. The 
third is to be similarly arrived at, by twisting the corners of a 
square having rounded corners ; the fourth, by twisting the corners 
of a regular pentagon having rounded ‘corners ; the fifth, by twisting 
the corners of a hexagon, and so on. 
In each of the annexed diagrams of toroidal helixes a circle is 
introduced to guide the judgment as to the relief above and 
depression below the plane of the diagram which the curve repre- 
sented in each case must be imagined to have. The circle may 
be imagined in each case to be the circular axis of a toroidal core 
on which the helix may be supposed to be wound. 
To avoid circumlocution, I have said, “give a right-handed 
twist ” in each case. The result in each case, as in fig. 1, illus- 
trates a vortex motion for which the corresponding rigid body 
describes left-handed helixes, by all its particles, round the central 
axis of the motion. If now, instead of right-handed twists to the 
plane of the oval, or the corners of the triangle, square, pentagon, 
&c., we give left-handed twists, as in figs. 2, 3, 4, the result in 
each case will be a vortex motion for which the corresponding 
rigid body describes right-handed helixes. It depends, of course, 
on the relation between the directions of the force resultant and 
couple resultant of the impulse, with no ambiguity in any case, 
whether the twists in the forms, and in the lines of motion of the 
corresponding rigid body, will be right-handed or left-handed. 
8. In each of these modes of motion the energy is a maximum- 
minimum for given force resultant and given couple resultant of 
impulse. The modes successively described above are successive 
solutions of the maximum-minimum problem of § 4; a determinate 
problem with the multiple solutions indicated above, but no other 
