The linearity of the governing equations enables slip line fields to be superimposed. This can be accomplished geometrically by using position vectors, or algebraically if slip lines are regarded as elements of an abstract linear vector space. The construction of complete fields can be represented formally by linear operators. The symmetry of the governing equations is reflected in some simple identities between these operators, which have far reaching geometrical consequences. The particular fields corresponding to the eigenvectors of these operators are investigated. The completeslip line field corresponding to a particular boundary value problem can be analysed in terms of these operators. By making use of the basic operational identities, relations can sometimes be established between distinct complete fields and between different parts of the same field. This considerably reduces the necessary computation. In addition, a novel computational procedure is developed by approximating the linear operators by finite-dimensional matrices.