  A brief critical review is given of the 50-year history of experimental and
theoretical studies of the magnetic properties of a new promising class of weak
ferrimagnets such as RFe$_{1-x}$Cr$_x$O$_3$ with a non-magnetic R-ion (R = La ,
Y, Lu), i.e. systems with competing signs of the Dzyaloshinsky vectors Fe-Fe,
Cr-Cr and Fe-Cr. The spin Hamiltonian of the system is considered taking into
account isotropic exchange, antisymmetric Dzyaloshinsky-Moriya exchange, and
second- and fourth-order single-ion anisotropy. Within the framework of the
molecular field approximation, calculations were made of the Neel temperatures,
the average magnetic moments of $3d$ ions, total and partial magnetizations,
and effective anisotropy constants. The existence in the model system
YFe$_{1-x}$Cr$_x$O$_3$ of two regions of negative magnetization $0.25 \leq
x\leq 0.5$ and $x\approx 0.8$ with the corresponding magnetization reversal
points reaching room temperature at $x\approx 0.45$. The phenomenon of spin
reorientation observed for single-crystal samples in a wide range of
concentrations is explained by a sharp decrease in the contribution of
antisymmetric exchange to magnetic anisotropy with increasing deviation from
the parent compositions and competition between the contributions of single-ion
anisotropy of Fe and Cr ions. It has been suggested that the spatial
orientation of the Neel ${\bf G}$ vector and the $G_{xyz}$ configuration are
the reason for the small value of saturation magnetization observed
experimentally for compositions inside or near the region of negative
magnetization.
