According to paragraph four, why do Holmes and Watson examine the walking stick closely?

According to the fourth paragraph, Holmes and Watson's interest in the walking stick was the clues it would give them as to its owner.

I have, at least, a well-polished, silver-plated coffee-pot in front of me," said he. "But, tell me, Watson, what do you make of our visitor's stick? Since we have been so unfortunate as to miss him and have no notion of his errand, this accidental souvenir becomes of importance. Let me hear you reconstruct the man by an examination of it."

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