Leary III seems to be a Labrador mix of some kind, and was named for an earlier Leary who died of undisclosed causes. There was, presumably, a Leary II.

Jake and Helen knew only the original Leary. Jake seems to have had a particularly good relationship with the original, but the current incarnation ignores him, and disgusts Helen with his filthy habits (no more filthy than any other dog, but still).