Major premise. All hounds are creatures that howl when they see strangers. (All M are P)
Minor premise. The hound at the mansion didn't howl when he saw the thief. (No M are S)
Conclusion. Therefore, the thief was not a stranger. (No S are P)
The syllogism is AEE in the third figure. This syllogism is invalid. If the major term (stranger) is distributed in the conclusion (it is), then it must be distributed in the premises (it is not).
The reasoning certainly sounds valid, but it does not write that way.
Added edit. These statements describe the line of reasoning:
Premise: If a hound sees a stranger, the hound howls.
Hypothetical assumption: The thief was a stranger and the hound saw him.
Hypothetical conclusion: Therefore, the hound howled. Modus ponens
Premise: But the hound did not howl when he saw whomever he saw.
Final conclusion: Therefore, the hound did not see a stranger. Modus tollens