  In special relativity, clock networks may be self-consistently synchronized
in an inertial frame by slowly transporting clocks, or by exchanging
electromagnetic signals between network nodes. However, clocks at rest in a
rotating coordinate system--such as on the surface of the rotating
earth--cannot be self-consistently synchronized by such processes, due to the
Sagnac effect. Discrepancies that arise are proportional to the area swept out
by a vector from the rotation axis to the portable clock or electromagnetic
pulse, projected onto a plane normal to the rotation axis. This raises the
question whether paths of minimal or extremal length can be found, for which
the Sagnac discrepancies are zero. This paper discusses the variational problem
of finding such "isosynchronous" paths on rotating discs and rotating spheres.
On a disc, the problem resembles the classical isoperimetric problem and the
paths turn out to be circular arcs. On a rotating sphere, however, between any
two endpoints there are an infinite number of extremal paths, described by
elliptic functions.
