This paper generalizes to the case of curved surfaces certain theorems of plane conformal geometry due to Kasner.2 The results obtained are: expressions for the invariants of horn angles of first and second order contact and the general right angle, and the first few conformal symmetry formulas. Conceivably the invariants and the formulas which Kasner found for the plane might have carried over with no change whatever to surfaces, but the fact proves to be more interesting than that. In the cases of lowest order there is actually no change, but for higher order the Gaussian curvature of the surface makes its appearance.