The Z-kernel which was first considered by Schiffer for plane regions is extended to arbitrary open Riemann surfaces for a number of significant subspaces of the space of square integrable harmonic differentials Γ h . The Z-kernel for each of the subspaces considered is expressed in terms of the principal functions. Thus if W is an open Riemann surface and p and q the L x principal functions of W with singularities Re 1/z and Im 1/z respectively, then the following result is proved. THEOREM.

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... differential dpdq* is an ί-kernel for the space Γ h . The Z-kernel and another kernel function called the kkernel by Schiffer are applied to the solution of some well known extremal problems on open Riemann surfaces.