Emily Arundell writes to Hercule Poirot because she believes she has been the victim of attempted murder. However, unfortunately this letter is delayed and when Poirot receives it, she has been dead for some time. Her doctor, who has lost his sense of smell, says that she died of liver problems she had had for many years.
Emily's companion Miss Lawson is the unexpected beneficiary of a substantial fortune, according to a very recent change of will. Under the previous will, Emily's nephew Charles Arundell and nieces Theresa Arundell and Bella Tanios would have inherited. This gives them all motive for murder, because it is unclear who knew of the changed will.
While examining the house, under a pretence of buying it, Poirot discovers a nail covered with varnish and a small string tied to it. Before her death Miss Arundell had said something about Bob...dog...picture...ajar. Poirot concludes that this means a jar on which there is a picture of a dog who was left out all night -- meaning that Bob could not have put the ball on the staircase because he had been out all night. Poirot concludes Miss Arundell had fallen over the tripwire.
On the day of her death Emily had been at a seance held by both Miss Tripps. Both Miss Tripps, two sisters who believe in seances, say that when Emily spoke, a luminous figure came from her mouth. Miss Lawson, who was also at the seance, similarly claims that a luminous haze appeared.
Theresa and Charles want to have the will contested and even offer to pay Poirot for it. Poirot seemingly agrees. He asks Bella, who, after talking with her husband, agrees. While at Emily's house Poirot talks to the gardener and finds out that Charles talked to him about his weed killer which turns out to be arsenic. The bottle is also nearly empty -- something that the gardener finds surprising.
Theresa Arundel is a strong suspect because Miss Lawson can recall seeing someone through her bedroom mirror at the top of the stairs on the night of Emily's accident. The person was wearing a brooch with the initials, "TA".
Bella leaves her husband, while saying he was trying to poison her, and goes to stay with Miss Lawson, but Poirot tells her to go to a certain hotel. The next day, she is found dead. A bottle of sleeping pills she had bought on the day Emily had been tripped up is in her hand. The murderer has apparently struck again.
Poirot discovers that Emily Arundell died of phosphorus poisoning, administered in her liver pills. The reason why haze appeared from her mouth was that her breath was phosphorescent. The reason her doctor did not know was because he could not smell the odour. The nature of the murder suggests a doctor. Dr. Donaldson, Theresa's fiancé, has a good motive for the crime, as does Bella Tanios's husband, also a doctor.
At a meeting with all the suspects, Poirot reveals that Theresa took the arsenic. However, she could not bear to take someone else's life, so she threw the arsenic away. The real murderer was Bella. She committed the murder for money to educate her children. Secretly, she had begun to hate her husband, and was planning to kill him as well. She killed herself because she knew that Poirot had begun to suspect her. The brooch that Miss Lawson had seen through the mirror was Bella's with the initials "AT" for Arabella Tanios; they appeared as "TA" because Miss Lawson was looking through the mirror. On her deathbed, Emily had asked Miss Lawson for the new will, presumably to destroy it, but Miss Lawson, thinking the will was only for a few thousand pounds, said that her lawyer had it. On discovering that the inheritance was much greater than she had imagined, she was racked with remorse.
Respecting the original will, Miss Lawson voluntarily shares the estate with Emily's other relations, including Bella's children. The dog Bob becomes Hastings' new pet.