75 
1913-14.] Path of Ray of Light in Rotating Solid. 
the radius of curvature of the projected curve is f CQS V'oM > The curve 
^ J 2o)( m 2 -1) 
itself follows from this condition and the fact that \Js is constant. 
Within the limits considered it is part of a circular helix. Its radius of 
curvature is / L— in a direction perpendicular to the axis of 
2o> cos yfr 0 (nA 2 — 1) 
rotation. This conclusion holds good whether or not we consider our 
rotating body to have also a translatory motion with regard to the axes 
of reference. 
Note on Mr Anderson’s Paper. By Sir Joseph Larmor, 
M.P., F.R.S. 
Mr E. M. Anderson’s elegant argument may be paraphrased as 
follows : — 
Let v he any coplanar velocity of the medium, and set E = v(/ul 2 — 1)/m 2 
and D = c/iul; then the time of passage of a ray restricted to any artificial 
path is 
f— — , approximately J- f ds - J— [e cos <f> . ds . 
J D + E cos <f> ^ J DJ D 2 J ^ 
Now 
J E cos <f>ds = ^ v cos (f>ds , 
where the integral expresses, for a complete circuit, the circulation of the 
medium in the sense introduced by Lord Kelvin into Hydrodynamics. 
Here a circuit can be completed by any unvaried return path. Now, in 
coplanar Kinematics, the circulation round the contour of any area is 
|2cod(area), where co is the velocity of differential rotation or the vorticity. 
When co is uniform, the time of passage of the ray is, for a ray in the plane 
of motion, 
S T = A (length) - CyL (area). 
Now, when the length is maintained constant, S (area) = 0 for all possible 
variations when the curve is a circular arc. Therefore, as Mr Anderson 
reasons, when the length also is allowed to vary 
S(area) = AS(length), 
where the value of A can be calculated from the circular form. A particular 
circular arc can then he selected which will make St vanish for all small 
variations of its form without restriction to constant length ; and this will 
be the path of an unconstrained ray. 
