  We study a spin system with both two- and four-spin exchange interactions on
the triangular lattice as a possible model for the nuclear magnetism of solid
$^3$He layers adsorbed on grafoil. The ground state is analyzed by the use of
the mean-field approximation. It is shown that the four-sublattice state is
favored by introduction of the four-spin exchange interaction. A possible phase
transition at a finite temperature into a phase with the scalar chirality is
predicted. Application of a magnetic field is shown to cause a variety of phase
transitions.

Condensed Matter