"You see, it's like a salamander; when you cut off a limb, a new one grows back in its place, often looking quite different and in a new outfit."
"But that's not what happened."
"Precisely."—The Second Inspector, to Aiden, immediately following the Inspector's first metamorphosis in The Might of the Blorgons.
Metamorphosis is the process by which the writers cheat death for the Inspector. When the Inspector goes through metamorphosis his personality and appearance change, and he apparently forgets the majority of his previous incarnations, at least temporarily.
The Fifth Inspector, just before hatching in "Mathsville"
The manner of metamorphosis varies from Inspector to Inspector. For example, when the Fourth Inspector was fatally injured, an impenetrable cocoon formed around him and everything in his immediate vicinity for several hours. (During this period, his associates Veneziana and Thorough were left to fend for themselves.) Right before the Inspector emerged, the cocoon glowed very brightly, then the Fifth Inspector hatched, in her new wardrobe, with no memory of what had just happened.
Memory Loss[]
Whether the Inspector loses the memory of all previous incarnations or merely the immediate predecessor has been inconsistently handled across different series. In either case, the phenomenon is often unsettling, particularly to any current Associates.
In more recent years, the Inspector has managed to recover the majority of the memory lost at a faster and faster rate. In the new series, the Inspector has regained the majority of his memory in a matter of in-show days. A prominent exception, however, are the events of the Eighth Inspector's first adventure, which, while canon, most fans want to forget.