Mr Satterthwaite is at a house party in the cold and rainy country, very much missing the comforts and warmth of his London home and looking forward to returning there, so much so that he has turned down an invitation from a young lady called Madge Keeley to join her and her father at his home at Laidell for another party starting that day. Falling asleep in front of the library fire, he is half-listening to a session of table-turning being conducted by the young members of the party in the same room. He is snapped wide-awake when the message spells out the word "QUIN" followed by "LAI". He instantly calls Madge Keeley and resurrects his invitation to Laidell.
Arriving at the house, he guesses from Madge's manner that her engagement is about to be announced and coming down to dinner soon after his arrival meets Roger Graham, her fiancé. Also there is Roger's mother and Madge's father, the owner of Laidell. David Keeley is an inconsequential man, little noticed by anyone despite being one of the most brilliant mathematicians of the age. At the dinner table there is also among the guests a young woman whose ethereal yet somewhat damaged beauty catches Satterthwaite's eye and he christens her "The Bird with the Broken Wing". He finds out that she is Mabelle Annesley, the youngest member of the Clydesley family, which has been dogged by ill-fortune over the years.
During the conversation between the men-folk that occurs after the ladies have left the table, Satterthwaite concludes that Roger Graham has something on his mind. On his way back to the party after briefly going to his room, Satterthwaite passes the terrace room and sees Mabelle inside it alone, illuminated by the moonlight. Talking to her he discovers that she saw a figure in the woods that day who could only have been Mr Quin. Taking this as a cue that she has some unhappiness in her life he finds out instead that she is blissfully happy. Mabelle's husband, Gerard, comes to fetch her to play as promised on her ukulele for the party and she does so, playing and singing for the assembled group. They then break up for the night and go to their respective rooms.
The next morning, Satterthwaite is woken up by a shocked Madge who breaks the news that Mabelle has been found dead, supposedly by suicide, hanging herself on the back of the door. The police arrive and the case is conducted by an old acquaintance of Satterthwaite – Inspector Winkfield – who addresses the assembled group to ascertain the facts. Mabelle had been prompted to return to drawing room to fetch her ukulele by Madge and that is the last that anyone saw of her. Gerard Annesley fell asleep in his adjoining room half an hour later without hearing her coming to bed and heard no sounds of her suicide in the night. Mr Satterthwaite speaks out that he is certain that the lady did not kill herself but in fact she was murdered. The Inspector questions him alone and confirms that his suspicions are correct. The rope which they found round her neck was much thicker than the cord marks that they found on her. Thinking through the events of the night and following morning, Satterthwaite remembers seeing smoke coming from Mrs Graham's room. She is a non-smoker and he investigates the cause. He finds some partially burnt letters in the grate which appear to be from Mabelle to Roger about an affair that they are conducting. He confronts Mrs Graham, who admits that she burnt them to stop her son being suspected of murder. Roger admits they were lovers but claims he didn't kill her.
Returning to the drawing room Satterthwaite finds one of the other house guests sadly holding Mabelle's ukulele in her hands. Strumming the strings, one snaps and they notice that it is an A-string but that it is in the wrong place. Satterthwaite suddenly realises that the murder was committed with the original string and confronts David Keeley in the library. He was the last person downstairs, switching off the lights as Mabelle returned for her instrument. He quickly killed her and then later in the night returned to move the body to her own room. Laughing insanely, Keeley admits the crime as Winkfield walks into the room.
Returning home by train, Satterthwaite meets Quin and sadly admits he failed to prevent Mabelle's death. Quin replies that he did save the wrong people being accused of the crime and asks if there are not greater evils than death. Satterthwaite closes his eyes, thinking about Mabelle, and when he opens his eyes, Mr Quin has already gone, but there's a bird done from a blue stone where Quin had been seated.