  The torsional oscillator is the chief instrument for investigating
supersolidity in solid 4He. These oscillators can be sensitive to the elastic
properties of the solid helium, which show anomalies over the same range of
temperature in which the supersolid phenomenon appears. In this report we
present a detailed study of the influence of the elastic properties of the
solid on the periods of torsional oscillators for the various designs that have
been commonly employed in supersolid measurements. We show how to design an
oscillator which measures supersolidity, and how to design one which
predominantly measures elasticity. We describe the use of multiple frequency
TOs for the separation of the elastic and supersolid phenomena.
