Polarised neutron diffraction measurements have been made on HoFeO$_3$ single crystals magnetised in both the [001] and [100] directions ($Pbnm$ setting). The polarisation dependencies of Bragg reflection intensities were measured both with a high field of H = 9 T parallel to [001] at T = 70 K and with the lower field H = 0.5 T parallel to [100] at T = 5, 15, 25~K. A Fourier projection of magnetization induced parallel to [001], made using the $hk0$ reflections measured in 9~T, indicates that almost all of it is due to alignment of Ho moments. Further analysis of the asymmetries of general reflections in these data showed that although, at 70~K, 9~T applied parallel to [001] hardly perturbs the antiferromagnetic order of the Fe sublattices, it induces significant antiferromagnetic order of the Ho sublattices in the $xmhyphen y$ plane, with the antiferromagnetic components of moment having the same order of magnitude as the induced ferromagnetic ones. Strong intensity asymmetries measured in the low temperature $Gamma_2$ structure with a lower field, 0.5 T $parallel$ [100] allowed the variation of the ordered components of the Ho and Fe moments to be followed. Their absolute orientations, in the 180degree domain stabilised by the field were determined relative to the distorted perovskite structure,. This relationship fixes the sign of the Dzyalshinski-Moriya (D-M) interaction which leads to the weak ferromagnetism. Our results indicate that the combination of strong y-axis anisotropy of the Ho moments and Ho-Fe exchange interactions breaks the centrosymmetry of the structure and could lead to ferroelectric polarization.
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