In this paper, a geometrical description of Lemke’s algorithm is presented for solving the linear complementarily problem: Find nonnegative vectors x and y satisfying $x = My + q$, $x^T y = 0$. This description is analogous to the simplicial description of the simplex method. A study is made of the class of all complementary cones and it is shown that, under a regularity condition on this class, necessary and sufficient conditions for the success of Lemke’s algorithm can be proved. It is also shown that this regularity condition is satisfied by some known classes of complementary cones. A class of matrices M for which all principal minors are negative is added to the existing classes of matrices for which the linear complementarily problem can be solved by Lemke’s algorithm.