Teichmüller's modulsatz and the variation of the Dirichlet integral

We show that changing the level curve of a harmonic function with the classical Hadamard variation with a small parameter entails a change in the Dirichlet integral of the function which is quadratic in the parameter. As a corollary, we supplement the well-known theorem of Teichmüller about the sum of moduli of doubly connected domains into which an annulus is subdivided by a continuum that differs little from a concentric circle.