The 180° rule at a class of higher-order triple points

The 180^∘{} rule restricting the disposition of phase boundaries at triple points is generalized to a class of triple points involving transitions of higher than first order. At the triple point considered, two Ehrenfest second-order transition lines meet a first-order line. The recently discovered "polycritical point" in ³He is an example of such a triple point. It is shown, subject to certain explicit assumptions, that none of the three phases may occupy more than 180^∘{} of angle at the triple point. Our assumptions are weaker and our results stronger and more general than those in a recent treatment of the rule by Leggett. We show that each of our assumptions is essential: If any one is relaxed, a counterexample to the 180^∘{} rule can be exhibited.