General solutions to the Stokes equations for a sphere held fixed in a general quadratic flow are used to develop the hydrodynamic effects of a thin layer of material at the surface which has different rheological properties than the solution. These surface-layer effects are developed as an expansion toO(λ2) where λ is the ratio of the length scale of the surface layer (δ) to the particle radius (R). The formalism is developed such that the force, couple and stresslet on the coated sphere are calculated directly by substituting a rheological model for the surface layer into an analysis based on unidirectional flow; thus, the hydrodynamic effects toO(λ2) can be determined without solving the Stokes equations outside the layer. TheO(λ2) effect is independent of the type of flow (translation, rotation, extension) in the sense that a single parameter A, which depends only on the properties of the surface layer, applies to all flow types. However, theO(λ2) effects depend both on the properties of the surface layer and the type of flow about the particle. Examples are presented as models for particles in solutions of nonadsorbing and adsorbing polymers. In the case of an adsorbing polymer whose layer is modeled by the Brinkman equation, theO(λ2) effects can be computed from the lowest order description of velocity field, that is, viscous flow past a flat surface. The hydrodynamic interactions between two particles and between a particle and a solid boundary are developed by a method of reflections accounting for the presence of the surface layers on the particles.