What are Joel Cairo's and Brigid O'Shaughnessy's involvement in the plot and what do they want Spade to discover?
Joel Cairo has been retained by the Russian General Kemidov to retrieve the Maltese falcon, which was stolen from him by Brigid O'Shaughnessy and brought to San Francisco from the Orient by her and Floyd Thursby. Cairo comes to Spade's office in the early evening. He has read in the papers that Spade's partner Miles Archer was killed and then Floyd Thursby was gunned down a few hours later. Cairo thinks Spade killed Thursby to avenge Archer and may have gotten possession of the falcon. Cairo pulls a gun on Spade and says he is going to search his office. But Spade takes the gun away, and Cairo ends up paying Spade a two-hundred-dollar retainer fee to get the falcon.
Brigid originally came to Spade's office earlier because she had a scheme to get rid of Thursby so that she wouldn't have to share the proceeds of the sale of the black statuette with him. Thursby was a mobster wanted by the police. Brigid thought she could get a private detective to tail him and Thursby would be scared into leaving town, thinking the other man was a cop or an enemy gangster. Either that, Brigid thought, or Floyd would get into a shoot-out with the private eye and either get killed or have to go on the lam to avoid being arrested for murder. Or if Floyd did get arrested, she would be rid of him that way. Miles Archer volunteers to tail Thursby that night. Brigid points Archer out to Thursby, but Thursby isn't sufficiently alarmed and doesn't try to shoot Archer. So Brigid separates from Thursby and leads Archer up a blind alley, where she kills the infatuated, unsuspecting private eye with one of Thursby's guns.
Brigid knows that Thursby was really killed by Caspar Gutman's gunman, Wilmer Cook. She is terrified because she knows she will be next. It was Gutman who hired her to steal the falcon from General Kemidov. She got it but double-crossed Gutman and kept the falcon. She gets in touch with Sam Spade because she needs someone to protect her after Thursby has been killed. Brigid doesn't have the falcon. She had Captain Jacobi bring it over on his own ship while she and Thursby sailed to San Francisco on a faster passenger liner. She is just waiting for Jacobi to bring her the black bird so that she can sell it. What she wants from Spade, besides protection, is for him to keep it a secret that she is his client, having come to his office to get a private detective to trail Thursby that night. Spade might have taken the job himself, but Miles Archer volunteered because he was strongly attracted to the beautiful young Brigid. That's why it was easy for her to lead him up the blind alley and kill him. In other words, Brigid now wants Spade to protect her from both the police and from Caspar Gutman. Once she had the statuette, she would probably do something to double-cross Spade and get rid of him; she might even be thinking of killing him.
Cairo wants Spade to get the falcon for him, but Brigid doesn't want Spade to "discover" anything. She just wants protection and secrecy until she can recover the black bird and get away from San Francisco. But Spade makes her confess that she killed his partner with the intention of pinning the murder on Floyd Thursby.
Spade said: "You thought Floyd would tackle him and one or the other of them would go down. If Thursby was the one then you were rid of him. If Miles was, then you could see that Floyd was caught and you'd be rid of him. That it?"
"S-something like that."
"And when you found that Thursby didn't mean to tackle him you borrowed the gun and did it yourself."