76 
Proceedings of the Royal Society of Edinburgh. [Sess. 
If an additional uniform velocity is imposed at right angles to this 
coplanar (or rather uniform laminar) motion, the vorticity will remain 
unaltered ; thus, in the expression for St (the ray being now free of 
any restriction to a plane) the area that occurs will be that of the 
projection on the plane of the laminar motion. Now, even in the most 
general coplanar motion, when the vorticity is not uniform, Jo)c£(area) 
will be stationary for small variations which leave the length unvaried, 
only when the curve is a geodesic on the cylinder, perpendicular to 
the laminar motion, on which it lies; for otherwise a displacement of 
the curve, considered as a thread, on the cylinder will make it slack, 
and the cylinder can be expanded to take in more area. This geodesic, 
as it would unroll into a straight line, will have the length of its projection 
on the plane of the laminar motion unvaried while the shape of the cylinder 
thus varies. The shape of the cross-section of the cylinder on which the 
ray lies will therefore be such that for given length of its arc Jwd( area) 
is stationary. In particular, if co is uniform, it will be an arc of a circle.* 
This brings us to Mr Anderson’s result, in a somewhat extended form. 
If a uniform material medium is in motion through the aether with 
vorticity co restricted to be constant in magnitude and direction, all rays 
of light travel in it along helices traced on cylinders of constant radius 
C fl 
2co ju 2 —l 
vorticity. 
* The same argument establishes that a flexible conductor carrying an electric current 
in a uniform magnetic field will when free assume the form of a circular helix ; ef. Proc. 
Lond. Math. Soc., vol. xvi., 1884, p. 169. 
cos'll, having their axes in the direction of the constant 
(Issued separately March 20, 1914.) 
