85 
The observed phenomena of rings, whether these be 
vortex rings or material rings which make space multiply 
continuous, are the effect of the cyclic motion. It is an 
object of this paper to notice the motion of the ring itself 
as the consequence of the cyclic motion. 
The energy of a cyclic motion may be written (Lamb’s 
Hydrodynamics, Art. 136) 
T = 2 J J f { u{y£ - zrj) + v{zl - x£) + w(xrf- yl) j dx dy dz 
and evidently depends on the size and shape of the necessary 
vortex filament. To find in what way a portion contributes 
to this energy by its motion, take the origin on a vortex 
line, the tangent line as axis of 0 , the principal normal as 
axis of x, and the binormal as that of y, and consider the 
part of the integral between s= —l and s=l. When we put 
s 3 s 2 s 3 
x = Xl + —y = y 1 + - e = s __ 
Xi and yi denoting the coordinates of a point in the section 
of the filament in the plane of xy, 
we obtain 
i 
Zdsda 
= 4U £ J y\Cda — 4V<? J x^da 
■/<*■ 
2U„ 
+ V 
when U and V are the mean resolved velocities of the ele- 
ment. 
If the core be of symmetrical section, the first two of 
these integrals will vanish, and the term contributed to the 
energy will contain only the velocity along the principal 
normal. 
The total energy will depend upon the size and shape, as 
said before ; but we can conclude that if the filament be 
of symmetrical section, the motion of each portion will be 
along its principal normal. In the case of a straight fila- 
