An elation of a design is an automorphism γ of fixing some block X pointwise and some point x on X blockwise. Luneburg  and I  have proved results which state that a design admitting many dations and having additional properties must be the design of points and hyperplanes of a finite desarguesian projective space. In this note, additional results of this type will be proved and applied to yield a generalization of a previous result on Jordan groups . The proofs were suggested by a result of Hering on dations of finite projective planes [1, pp. 122, 190].