Note on the Location of the Critical Points of Harmonic Functions.

THEOREM 1. Let the region R of the extended (x, y)-plane be bounded by a finite number of mutually disjoint Jordan curves Co, G., C2, • • • , Cn. Let the function u(x, y) be harmonic in R, continuous in the corresponding closed region, equal to zero on Co and to unity on G, C2, • • • , Cn. Denote by Ro the region bounded by Co containing the curves Ci, C2, • • • , Cn in its interior; define noneuclidean straight lines in Ro as the images of arcs of circles orthogonal to the unit circle, when Ro is mapped conformally onto the interior of the unit circle. If II is any non-euclidean convex region in R0 which contains all the curves Ci, C2> * • * » Cn» then II also contains all critical points of u(x, y) in R.