On Weyl's Gauge Field

The relationship between the scale transformation and Weyl’s gauge transformation is investigated. It is shown that a scale invariant Lagrangian can be transformed into a scalar density which is invariant under x-dependent gauge transformations in Weyl’s sense. The Lagrangian of the latter form gives a symmetric energy momentum tensor the trace of which can be shown to vanish provided that some equation for Weyl’s gauge field is satisfied in addition to the equations of the original scale invariant fields. A simple example is investigated to show an extraordinary property of Weyl’s gauge field.