  Supersymmetry is used to derive conditions on higher derivative terms in the
effective action of type IIB supergravity. Using these conditions, we are able
to prove earlier conjectures that certain modular invariant interactions of
order alpha' **3 relative to the Einstein-Hilbert term are proportional to
eigenfunctions of the Laplace operator on the fundamental domain of SL(2,Z). We
also discuss how these arguments generalize to terms of higher order in alpha',
as well as to compactifications of supergravity.
