The Reactance Theorem for a Resonator

The Foster reactance theorem which states that in any loss-free network, dX/dω is positive, is here proved for any loss-free resonator. However, to establish the existence of an input impedance, the author feeds the resonator with a coaxial (or other suitable) transmission line. The proof is based upon an extension of Helmholtz´s theorem of adiabatic invariants. The variation of frequency is attained by a slow (adiabatic) movement of a short-circuiting plug in the transmission line while the cavity is oscillating.