Besides the five regular solids, there are other polyhedra which satisfy some, but not all, the conditions of regularity as usually defined, and so may properly be termed semi-regular. Some of these, which have an interest both as geometrical figures and as naturally connected with the geometrical problems of the partition of space and of crystallography, I propose here briefly to discuss.

We have seen that for solving problems on the simultaneous impact of more than two bodies, we require in addition to the laws of Abstract Dynamics, not only a generalization of Newton’s Empirical Law as to change of relative velocity, but also the assumption that the impacts are simultaneous in the special sense of having their greatest compression at the same instant. In other words, it is only in a particular set of special cases that elementary methods can lead to a solution.