Adjoint Transform of Willmore Surfaces in $n$-sphere

After the surface theory of M\"obius geometry, this study concerns a pair of conformally immersed surfaces in $n$-sphere. Two new invariants $\theta$ and $\rho$ associated with them are introduced as well as the notion of touch and co-touch. This approach is helpful in research about transforms of certain surface classes. As an application, we define adjoint transform for any given Willmore surface in $n$-sphere. It always exists locally (yet not unique in general) and generalizes known duality theorems of Willmore surfaces. This theory on surface pairs reaches its high point by a characterization of adjoint Willmore surfaces in terms of harmonic maps.