The 39 Steps

At a London music hall theatre, Richard Hannay (Robert Donat) is watching a demonstration of the superlative powers of recall of "Mr. Memory" (Wylie Watson) when shots are fired.[1] In the ensuing panic, Hannay finds himself holding a seemingly frightened Annabella Smith (Lucie Mannheim), who talks him into taking her back to his flat. There, she tells him that she is a spy, being chased by assassins, and that she has uncovered a plot to steal vital British military information, masterminded by a man with the top joint missing from one of his fingers. She mentions the "39 Steps", but does not explain its meaning.
Later that night Smith, fatally stabbed, bursts into Hannay's bedroom and warns him to flee. He finds a map of the Scottish Highlands clutched in her hand, showing the area around Killin, with a house or farm named "Alt-na-Shellach" circled. He sneaks out of the watched flat disguised as a milkman and boards the Flying Scotsman express train to Scotland. He learns from a newspaper that he is the target of a nationwide manhunt for Smith's murderer. When he sees the police searching the train, he enters a compartment and kisses the sole occupant, Pamela (Madeleine Carroll), in a desperate attempt to escape detection. She frees herself from his unwanted embrace and alerts the policemen, who stop the train on the Forth Bridge. Hannay escapes, however.
He walks toward Alt-na-Shellach, staying the night in the house of a poor crofter (John Laurie) and his much younger wife (Peggy Ashcroft). Early the next morning, she sees a police car approaching and warns Hannay. Hannay flees, wearing the crofter's coat. At a bridge, he finds a sign for Alt-na-Shellach. He arrives at the house of the seemingly respectable Professor Jordan (Godfrey Tearle) and is let in after saying he has been sent by Annabella Smith. The police arrive, but Jordan sends them away and listens to Hannay's story. Jordan then reveals that he is missing part of a finger; he shoots Hannay and leaves him for dead.
Luckily, the bullet is stopped by the crofter's hymn book in the coat pocket. Hannay drives into town and goes to the sheriff, who does not believe the fugitive's story since he knows Jordan well. Hannay's right wrist is handcuffed, but he jumps through a window and escapes by joining a Salvation Army march through the town. He tries to hide at a political meeting and is mistaken for the introductory speaker. He gives a rousing impromptu speech â without knowing anything about the candidate he is introducing â but is recognised by Pamela, who gives him up once more. He is taken away by "policemen" who ask Pamela to accompany them. They drive past the police station, claiming they have orders to go directly to Inveraray, but Hannay realises they are agents of the conspiracy when they take the wrong road. When the men get out to disperse a flock of sheep blocking the road, Hannay escapes, dragging the unwilling Pamela (to whom he is handcuffed) along.
They make their way across the countryside and stay the night at an inn. While he sleeps, Pamela manages to slip out of the handcuffs, but then overhears one of the fake policemen on the telephone, confirming Hannay's assertions. She returns to the room and sleeps on a sofa. The next morning, she tells him what she heard. He sends her to London to alert the police. No secret documents have been reported missing, however, so they do not believe her. Instead, they follow her.
Pamela leads them to the London Palladium. When Mr. Memory is introduced, Hannay recognises his theme musicâthe annoyingly catchy tune he has been unable to forget for days. Hannay realises that the spies are using Mr. Memory to smuggle the secrets out. As the police take him into custody, he shouts, "What are the 39 Steps?" Mr. Memory compulsively answers, "The 39 Steps is an organisation of spies, collecting information on behalf of the Foreign Office of ..." Jordan shoots him and tries to flee, but is apprehended. The dying Mr. Memory recites the information stored in his brainâthe design for a silent aircraft engineâand is then able to pass away peacefully, saying "I'm glad it's off my mind."
The film fades to an image of Hannay and Pamela's clasped hands as they stand at the side of the stage while the hurriedly ushered-on chorus line dance to an orchestrated version of the Jessie Matthews song "Tinkle Tinkle Tinkle".
(Hitchcock had worked with Jessie Matthews on the film Waltzes from Vienna and reportedly did not like her very much, but as well as the fade-out music to "The 39 Steps", he also used an orchestrated version of her song "May I Have The Next Romance With You" in the ballroom sequence of his film Young and Innocent.)

Question: What Jessie Matthew's song was playing?
Answer: Tinkle Tinkle Tinkle