Sphinx begins, as does The Haunted Woman, with its young central character inheriting enough money to set him up for a life without needing to work. But, whereas Isbel Loment had not, by the end of her book, succeeded in breaking away from her old life into the new one her inheritance would enable her to live, Sphinx's Nicholas Cabot has already started on his.
He arrives at the village of Newleigh, where he is to rent a room at the Sturt household, intending to devote himself to his particular line of scientific enquiry: the invention of a machine to record deep-sleep dreams.
Sturt, a retired actor, lives with his wife and three daughters, Audrey, Evelyn and Katherine, aged 19, 21 and 23. Audrey is flighty and interested in tennis, Evelyn is serious and musical (a sure sign from Lindsay that she is important), and Katherine, though literary, is more interested in belles-lettres than intellectual matters. Their neighbour, Celia Hantish, is a fashionable young widow.
Nicholas soon learns of another neighbour, the composer Lore Jensen. Lore's early compositions showed the promise of genius, but she was forced to turn to more commercial work to earn a living. She now has the money she needs to live, but her inspiration has dried up, and she has turned to drink and drugs. Evelyn plays him an early piece of Lore's entitled Sphinx. It seems to be posing an "indubitable question". Katherine explains: "the Sphinx was a monster that used to ask riddles... those who were unable to solve them correctly lost their lives." Evelyn thinks the riddle is, "'Why are you living in the world?' As none of us can answer it, we all have to die." Nicholas disagrees: for him the Sphinx is the goddess of the "dreams we dream during deep sleep and remember nothing of afterwards."
The story that follows is convoluted. Maurice Ferreira, a suave, selfish philanderer that Nicholas has employed for a short time, is carrying on affairs with both Evelyn and Lore. Things come to a head when, drunk, he produces a revolver and holds it to Lore's head, demanding that she marry him. Rumour of the event gets round, and is taken by Mrs Sturt and Katherine to be evidence of a dissolute lifestyle on Lore's part; they break off relations with her. However, Sturt is (unbeknown to his wife) Lore's father. But, because of Lore and Sturt's obviously close relationship, the ignorant Ferreira threatens to spread rumours that these two are having an affair, as a way of getting back at Lore when she refuses to see him. Nicholas, who has been acting as a general go-between, gets into a scuffle with Ferreira and relieves him of his revolver.
Celia Hantish (the widow neighbour) reconciles the Sturts and Lore just so as to fend off scandal. Things start to return to normality. Then Lore announces she is to marry a music critic, Monro Dawson. Ferreira has now finished his work for Nicholas, and has his revolver back. He declares his intention to gatecrash a fête to be given by Lady Wyburn, at which Lore and Dawson will be present.
Up to this point, Nicholas has been making sporadic progress with his dream-recording device. He has managed to record three dream-fragments, all of which he plays to Evelyn (the only other character to know the object of his research.) In the first dream, Nicholas meets Lore on a forest path. Her expression is one of "agonised despair". She begs him to help her. (After this, Evelyn insists they go round to Lore's house to try, but this is just after the snub by Mrs Sturt, and they are turned away). In the second recorded dream, Lore is seen walking towards Ferreira, who has the face of a devil. In the third (viewed once Ferreira has been dismissed) it becomes evident that, in the dream-world, Ferreira has killed Lore.
In the meantime, Nicholas has become infatuated with the sophisticated Celia Hantish and she has agreed to marry him. He doesn't realise they are hopelessly mismatched, even when she shows obvious disappointment at his intention to continue his work after they're married.
The day of the fête arrives. Nicholas and Celia meet Lore and her fiancé Dawson, who have just been arguing. Dawson leaves and Lore says the engagement is off. As Nicholas and Celia walk away, Lore stares into a river, depressed, and Nicholas sees someone he guesses to be Ferreira approaching her. Soon after, the fête is called to a halt; Lore has been found in the river, drowned.
Celia Hantish tries to get Nicholas to place the blame on Ferreira, but he refuses to swear it was Ferreira he saw. Seeing it as a personal affront, she breaks off their engagement.
Back home, Sturt collapses on hearing of Lore's fate. Nicholas, knowing he will be asked to provide evidence, decides to record a dream and see what it will reveal of Ferreira's involvement. Evelyn, who has an obscure desire to protect Ferreira, doubles Nicholas's dose of sleeping powder, and puts the dream recorder in her father's room, hoping it will record an innocent dream which will indict nobody.
She watches it as soon as it is recorded. It provides a sort of three-level interpretation of recent events. In the dream, her father, Sturt, is walking along a beach. He stops at a pool. Running under the surface of the water is a sort of air-tunnel, and Lore is trapped in it. Below are a series of shadow-images of Lore which present different views of her predicament. In one shadow-image, she is walking along a forest path — this is the Lore that Nicholas met in his dreams. Below that is another shadow-image of Lore, in which she is standing by a river's edge — this shadow level is the everyday waking world. In it, Ferreira appears, threatens Lore with his revolver but leaves, whereupon Lore throws herself into the river. In the "forest" shadow-image, however, Ferreira murders Lore. At "water-tunnel" level, he appears not as a person but as a dark tunnel-orifice. Confronted with this, Lore reaches up and climbs out of the pool. So, in one interpretation, Ferreira has killed Lore; in another, he was merely the conduit by which she freed herself. In the main dream world (on the beach), two white horses appear, one ridden by Nicholas. Lore mounts the other horse, and the two gallop towards the sea. The horses start to fly, and the dream ends.
Evelyn rushes to Nicholas's room, and finds that the extra dose of sleeping powder has killed him.