  We describe here the properties expected of a higher (with emphasis on the
order fourth) order phase transition. The order is identified in the sense
first noted by Ehrenfest, namely in terms of the temperature dependence of the
ordered state free energy near the phase boundary. We have derived an equation
for the phase boundary in terms of the discontinuities in thermodynamic
observables, developed a Ginzburg-Landau free energy and studied the
thermodynamic and magnetic properties. We also discuss the current status of
experiments on $Ba_{0.6}K_{0.4}BiO_3$ and other $BiO_3$ based superconductors,
the expectations for parameters and examine alternative explanations of the
experimental results.
