  We present a method to identify the coupled, normal modes of a
superconducting transmission-line with an embedded lumped element circuit. We
evaluate the effective transmission-line non-linearities in the case of
Kerr-like Josephson interactions in the circuit and in the case where the
embedded circuit constitutes a qubit degree of freedom, which is Rabi coupled
to the field in the transmission-line. Our theory quantitatively accounts for
the very high and positive Kerr non-linearities observed in a recent experiment
[M. Reh\'ak et.al., Appl. Phys. Lett. 104, 162604], and we can evaluate the
accomplishments of modified versions of the experimental circuit.
