The Zeeman splitting of nuclear quadrupole resonances is discussed and a formula given for the split resonance frequencies as a function of the angle between the perturbing magnetic field and the symmetry axis of the crystalline electric field. The direction of this axis in the crystal can be found If the electric field does not have cylindrical symmetry then for certain angles the spectrum becomes simplified and the directions of the three principal axes of the electric field gradient tensor can be found as well as the degree of asymmetry of the electric field. These resonances are observed with the aid of a super-regenerative oscillator. A brief description of its operation is given as well as some signal to noise ratio considerations for various methods of detection of the resonances. The spectrometer used is described. The Zeeman spectra of the nuclear quadrupole resonances of Br⁸¹ in single crystals of Zn(BrO₃)₂ .6H₂O and Co(BrO₃)₂.6H₂O are observed. The accuracy of the observations is discussed and the conclusion is reached that within the error of the experiment the crystalline electric fields have cylindrical symmetry with four different directions of the symmetry axes in the crystal. They are parallel to the{1,1,1} crystal axes. The crystals have cubic structure. / Science, Faculty of / Physics and Astronomy, Department of / Graduate