of Edinburgh, Session 1875-76. 63 
solution, when the vorticity is given in a single simple ring of the 
liquid. 
9. The problem of steady motion, for the case of a vortex line 
with infinitely thin core, bears a close analogy to the following 
purely geometrical problem : — 
Find the curve whose length shall be a minimum with given 
resultant projeetional area, and given resultant areal moment (§ 27 
below). This would be identical with the vortex problem if the 
energy of an infinitely thin vortex ring of given volume and given 
cyclic constant were a function simply of its apertural circum- 
ference. The geometrical problem clearly has multiple solutions 
answering precisely to the solutions of the vortex problem. 
10. The very high modes of solution are clearly very nearly 
identical for the two problems (infinitely high modes identical) 
and are found thus : — 
Take the solution derived in the manner explained above, from 
a regular polygon of N sides, when N is a very great number. It 
is obvious that either problem must lead to a form of curve like 
that of a long regular spiral spring of the ordinary kind bent round 
till its two ends meet, and then having its ends properly cut and 
joined so as to give a continuous endless helix with axis a circle 
(instead of the ordinary straight line-axis), and N turns of the 
spiral round its circular axis. This curve I call a toroidal helix, 
because it lies on a toroid * just as the common regular helix lies 
* 1 can a circular toroid a simple ring generated by the revolution of any 
singly-circumferential closed plane curve round any axis in its plane not 
cutting it. A “tore,” following French usage, is a ring generated by the 
revolution of a circle round any line in its plane not cutting it. Any simple 
