Lord Peter, cheerfully preparing to go bid for old books at an auction, gets a call from his mother sending him instead to view a dead body that was mysteriously found in a flat's bathtub.
The body's rather mysterious itself. When later, he meets his friend Inspector Charles Parker -- whose connection with the case was that he had checked to ensure it wasn't the missing person case he was on -- he recounts the anomalies about it, enough for Parker to realize it really had a good number.
Parker tells him about his own case, the famous financier Sir Reuben Levy have vanished from his home in the dead of the night. Lord Peter helps investigate that too.
It involves the Dowager Duchess's having known Lady Levy when they were girls, an advertisement for an owner of eyeglasses, a raid on a nightclub (though that's not on stage), a medical student remembering what he did one day, and much more.
It's obviously the first novel in one respect. Lord Peter took up detecting basically as a pastime, and it shows. Which, in due course, leads to his discussing it with Parker, who lays down the law that detecting murderers is not a game.