We present a progress report on our investigation of finite size effects on the superfluid transition. In our experiment, the helium is confined between the parallel plates of a carefully constructed capacitor. We measure the dielectric constant as a function of temperature, from which we can calculate the thermal expansivity. We discuss some of the following important features of our experiment: The capacitor, built in collaboration with the Chex experiment, gives us a simple, well defined planar geometry, extremely flat surfaces, and a sub-micron gap. A low temperature audio-frequency preamp improves the sensitivity of the capacitance measurement. A high resolution paramagnetic salt thermometer, also from Chex, enables us to work extremely close to the lambda point. We also present a novel design for a simple low temperature burst disc that is part of our apparatus.