The van der Pauw method is a well-known experimental techniquein the applied sciences for measuring physical quantities such as the electricalconductivity or the Hall coefficient of a given sample. Its popularity isattributable to its flexibility: the same method works for planar samples ofany shape provided they are simply connected. Mathematically, the method isbased on the cross-ratio identity. Much recent work has been done by appliedscientists attempting to extend the van der Pauw method to samples withholes (“holey samples”). In this article we show the relevance of two newfunction theoretic ingredients to this area of application: the prime functionassociated with the Schottky double of a multiply connected planar domainand the Fay trisecant identity involving that prime function. We focus hereon the single-hole (doubly connected, or genus one) case. Using these newtheoretical ingredients we are able to prove several mathematical conjecturesput forward in the applied science literature.