  We establish the existence of static doubly periodic patterns (in particular
rolls, rectangles and hexagons) on the free surface of a ferrofluid near onset
of the Rosensweig instability, assuming a general (nonlinear) magnetisation
law. A novel formulation of the ferrohydrostatic equations in terms of
Dirichlet- Neumann operators for nonlinear elliptic boundary- value problems is
presented. We demonstrate the analyticity of these operators in suitable
function spaces and solve the ferrohydrostatic problem using an analytic
version of Crandall-Rabinowitz local bifurcation theory. Criteria are derived
for the bifurcations to be sub-, super- or transcritical with respect to a
dimensionless physical parameter.
