  We derive an explicit matrix representation for the Hamiltonian of the Ising
model in mutually orthogonal external magnetic fields, using as basis the
eigenstates of a system of non-interacting \mbox{spin~$1/2$} particles in
external magnetic fields. We subsequently apply our results to obtain an
analytical expression for the ground state energy per spin, to the fourth order
in the exchange integral, for the Ising model in perpendicular external fields.
