71 
1913 - 14 .] Path of Ray of Light in Rotating Solid. 
calculus of variations, that for any given length of line joining two points 
A and B (fig. 2), that form of curve which subtends a maximum area with a 
third point O in the same plane is an arc of a circle. We shall not, however, 
use this result, but another which follows from it, or may be regarded as 
the last step in its demonstration, namely, that if AFDGB be the circular 
arc of required length, and ACDEB a slight variation from it having 
exactly the same length, then the areas 0 ACDEB and 0 AFDGB are 
equal to the first order of small quantities. 
If now (fig. 3) we consider ALB as a possible circular path for light 
in the rotating solid, and let APB be any slight deviation from it, in the 
same plane, but not necessarily of the same length as ALB ; then if the 
lengths of the two curves be not the same, let AMB be the circular 
arc joining A and B which has the same length as APB. Then by the 
theorem just stated AMB and APB subtend the same area at any point in 
their plane to the first order of small quantities. Thus the time taken by 
an ethereal disturbance to traverse the paths AMB and APB will be equal 
with the same degree of accuracy. 
Let us next consider the difference of length, and of area subtended by 
the two circular arcs ALB and AMB. Let R be the centre of the chord 
AB, and C and D the respective centres of the two circles. If then 
Ah = a 
AT) = p 
AC = p + e, 
then the area enclosed by the two arcs is 
