An analytical solution is presented for determining the optimum location of a radiating source on the surface of a sphere, given multiple bearings. The bearings are assumed to have small errors of the order of 0¿--10¿. The optimum location is found by minimizing the sum of the squares of the perpendicular great-circle distances from the source to the bearing lines. This is achieved analytically through an eigenvalue approach, rather than the usual iterative, numerical approach. Bearings of different weight are taken into account by approximating the distance from each direction finder to the source. The result is general and may have wide application. Since it is simple and nearly as fast as the triangulation technique for source location, it is now used in the SUNY-Albany East Coast Lightning Detection Network to compute the optimum location for lightning in real time. The solution is also used off-line to evaluate the systematic errors in the network direction finders. ¿American Geophysical Union 1987