  An exact analytical derivation is presented, showing that the Ising model on
the Cayley tree exhibits a line of third order phase transition points, between
temperatures $T_2=2k_B^{-1}J\ln ({\sqrt 2}+1) $ and $T_{BP}=k_B^{-1}J\ln (3)$,
and a line of fourth order phase transitions between $T_{BP}$ and $\infty$,
where $k_B$ is the Boltzmann constant, and $J$ is the nearest-neighbor
interaction parameter.
