On the day of Tulkinghorn's funeral, almost no one comes in person. However, most of his clients each send a carriage to indicate that they're in mourning. (Nice custom, right? No reason to actually show up when you can just send an empty carriage.)
Bucket rides hidden in one of the supposedly empty carriages and checks out the scene.
His wife, who is a pretty good amateur detective, hangs out on the steps of Tulkinghorn's house with her new tenant, all according to Bucket's instructions. We don't really know what the deal with that is.
After the funeral, Bucket heads to the Dedlock mansion. He's got his own key, and his own expense account, all on orders from Sir Dedlock.
The footman delivers Bucket a letter. It's just like a bunch that he's recently gotten – totally blank except for two words: "LADY DEDLOCK."
After dinner, he goes to fill Sir Dedlock in on what's going on.
Not too much news to report today, though.
Sir Dedlock makes a speech about how totally dedicated he is to finding Tulkinghorn's killer, and how even if his own brother did it, he'd want him hanged.
Bucket listens politely, then says he's very close to cracking the case wide open.
Then he goes to chat more with the footman.
Bucket does kind of a sneaky interrogation.
First he talks about how the footman is so tall and well-built that he should go be an artist's model. Then he asks about Lady Dedlock's walking habits. Oh, she likes to go out for walks, does she? How about that. Then he talks about how his dad was a footman and how proud he was about it. So was she out for a walk the night Tulkinghorn was murdered? Oh, she was? Very interesting. Then he talks about how good-looking she is. The footman is into it. Oh, and did you go with her for that nighttime walk? No? Well, good then.