The motion of a semi-infinite gas cloud bordering upon a vacuum is studied by the method of characteristics. Problems previously discussed by Burgers and Copson are illustrated by diagrams; they involve gas flows which are always continuous. It is shown that unless certain conditions are satisfied by the initial state in the cloud the subsequent flow cannot remain indefinitely continuous; shock-waves must appear within the gas. An example of the breakdown of continuity is given. A singular solution is considered in which all the characteristics of one family are concurrent in a point of the (x, t) plane. Finally, it is shown that the high eventual velocity of these gas clouds, which was considered to be a physically unsatisfactory feature of the previous solutions, is not to be avoided by variation of the initial inhomogeneities.