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While M. Bouc is now certain of the Countess's guilt, Poirot still has more detective work to do. In fact, he believes the Count when he says his wife did not murder the man. And then there's the handkerchief.
Speaking of which, Princess Dragomiroff bursts up into the room and informs the men that the handkerchief is hers. How? Her Christian name is Natalia, which starts with "N." "H" is an "N" in Russian.
She then says she has no idea how it got in there. She says she did not tell Poirot that the Countess was Mrs. Armstrong's sister out of loyalty.
She says that in this case "justice – strict justice – has been done" (3.5.38).
She can prove the handkerchief is hers by writing to the makers in Paris.
The maid hesitated when she saw the handkerchief, too, but protected the Princess out of loyalty.
The Princess leaves. Poirot thinks her power is "in her will rather than in her arm" (3.5.54).
Lies, lies, and more lies. And Poirot sets out to uncover more. How? By confronting people with his guesses and seeing what happens. When confronted with the truth, people usually admit it, if only out of shock.