  We prove that if (C,0) is a reduced curve germ on a rational surface
singularity (X,0) then its delta invariant can be recovered by a concrete
expression associated with the embedded topological type of the pair (X,C).
Furthermore, we also identify it with another (a priori) embedded analytic
invariant, which is motivated by the theory of adjoint ideals. Finally, we
connect our formulae with the local correction term at singular points of the
global Riemann--Roch formula, valid for projective normal surfaces, introduced
by Blache.
