The van der Pauw method is commonly used in the applied sciences to find the resistivity of a simply connected, two-dimensional conducting laminate. Given the usefulness of this ‘4-point probe’ method there has been much recent interest in trying to extend it to holey, that is, multiply connected, samples. This paper introduces two new mathematical tools to this area of investigation—the prime function on the Schottky double of a planar domain and the Fay trisecant identity—and uses them to show how the van der Pauw method can be extended to find the resistivity of a sample with a hole. We show that an integrated form of the Fay trisecant identity provides valuable information concerning the appearance of ‘envelopes’ observed in the case of holey samples by previous authors. We find explicit formulae for these envelopes, as well as an approximate formula relating two pairs of resistance measurements to the sample resistivity that is expected to be valid when the hole is sufficiently small and not too close to the outer boundary. We describe how these new mathematical tools have enabled us to prove certain conjectures recently made in the engineering literature.

van der Pauw方法在应用科学中通常用于查找简单连接的二维导电层压板的电阻率。鉴于这种“四点探针”方法的有用性，最近人们一直在试图将其扩展到有孔的（即多重连接的）样本。本文为这一研究领域介绍了两个新的数学工具-平面域的肖特基双上的素函数和Fay三正割恒等式-并使用它们展示了可以如何扩展van der Pauw方法来找到a的电阻率带孔的样品。我们表明，费伊三正割身份的综合形式提供了有关以前作者在有孔样品的情况下观察到的“信封”外观的有价值的信息。我们为这些信封找到了明确的公式，以及将两对电阻测量值与样品电阻率相关联的近似公式，当孔足够小且不太靠近外边界时，该公式有望有效。我们描述了这些新的数学工具如何使我们能够证明最近在工程文献中所做的某些猜想。