73 
1913-14.] Path of Ray of Light in Rotating Solid. 
The condition for a brachistochrone is 
ds 
D + E?' sin $ + F cos 0 
or to the first order of small quantities 
= 0 
fs-‘j 
dsFr sin <£ dsF cos 0 
D 2 
D 2 
= 0 . 
Now, as the projection of the path AB on a line drawn parallel to v 
is a constant, the last term vanishes, and we therefore arrive at our 
previous result. 
In both this and the previous case the curves calculated are those 
followed in space with regard to the assumed axes of no velocity. The 
paths traced out in the solid itself can be deduced as follows. It is easy 
to show that a ray which penetrates the rotating body in a straight line, 
with velocity - , will leave a trace in the solid of curvature } which will 
fk c 
he convex to the centre of rotation if the ray be moving with regard 
to the centre in the direction of rotation. The actual curvature of the 
o / 2 1 \ 
path of light is — — in the opposite direction. By subtraction we 
Cfk 
get — as the curvature of the path traced out in the solid ; the radius of 
Cfk 
curvature is ~ , and the curve will be convex or concave to the centre 
2oo 
of rotation according as the radius vector of the disturbance moves along 
with or against the rotation. This result will apply to both the cases 
so far considered. 
We have next to deal with the case in which the path of the ray is not in 
the plane of rotation. Let us consider what path will be followed between 
two points A and B in a medium rotating about an axis LM (fig. 4). Let 
P be the plane passing through A and perpendicular to LM ; 0 the point 
