Report discussing discrete energy levels observed within certain crystals which are due to perturbations of energy levels of the free ion by an electrostatic field arising from the crystal lattice. The analytic procedures for determining the field from the charge configuration are given, and the resulting fields are classified according to their symmetry. After a general survey of group-theoretical ideas, the applicable groups are analyzed in detail, and characters appropriate for both integral and half-integral angular momenta of the free ion are tabulated. Text includes tabulations, equations, and matrices using Wigner and Racah coefficients.