Laelaps was a hound destined to catch whatever it chased, whilst the Teumessian Fox was destined to never be caught.
The chase went on but Laelaps could not close on the Teumessian Fox, but nor could the Teumessian Fox get clear of the hound. Zeus looked down from Mount Olympus and observed the paradox that was taking place around Thebes. Zeus decided that he could not let the chase continue on earth, and so as the fox and hound ran over the plain of Thebes, the god changed both animals into stone.
Zeus then transformed the pair into two constellations, Canis Major (Laelaps) and Canis Minor (the Teumessian Fox), and so the chase continues in the night sky.