72 
Proceedings of the Royal Society of Edinburgh. [Sess. 
which equals 
2e( p sin _1 — 
V P Jo 2 - aV 
P Jp 2 - a 2 ' 
The difference in length of the two arcs is 
the former result divided by p, where p is the radius of curvature. 
Thus the area enclosed between two nearly coincident circular arcs, 
ending at the same points, is p times their difference of length. Also, from 
what we have already seen, the difference in the area subtended at any 
point in its plane by a circular arc and any slight deviation from it will 
be p times the difference in length of the two curves. This assumes that 
the point of subtension is so situated that the area swept out by the radius 
vector is wholly positive or wholly negative. Obviously, for a point on 
the concave side the difference of area will be of the same sign as the 
difference in length, while for a point on the convex side the sign will be 
opposite. 
Now, we have seen that the condition for a brachistochrone in the case 
we are considering is that an increase of length in the curve between any 
two points shall be accompanied by an increase of area subtended at the 
centre of rotation of times the amount, area being counted 
2co( J u 2 — 1) 6 
positive when traced out in the same direction as the body is rotating. 
This condition is obviously satisfied by a circular arc whose radius of 
curvature 
c p, 
W 2(p? - 1) 
and which is concave or convex to the centre of rotation according as the 
wave motion with regard to that centre is in the same or in the opposite 
direction to the rotation. 
We will next consider the case where the medium, in addition to its 
rotatory motion, has an uniform translatory motion v, as before, small com- 
pared to the velocity of light. If 6 be the angle between the direction 
of the ray and that of v, the total velocity is 
c <or sin <jf>(/x 2 - 1) fcos0(/x 2 -l) 
~ + ^2 + ^2 ’ 
where 
D + Er sin </> + F cos 0 , 
tr _ v ( A*- 2 ~ 1 ) 
