"Surely the Egyptian Sphinx was a personification of Nature?" she said, not looking at her sister, but at Nicholas. "The question which she asks, and which no one can answer, is, 'Why are you living in the world?' As none of us can answer it, we all have to die."— Sphinx, ch. II, "A Family Party"
Lindsay began writing Sphinx in August 1921, and, according to J B Pick ("A Sketch of Lindsay's Life as Man and Writer" in The Strange Genius of David Lindsay), found the going hard. He gave up on it at one point, then restarted in October of the same year. The first draft was completed in March 1922, but Lindsay continued to revise it till May.
Even having finished it, Lindsay's Sphinx continued to prove problematic. Methuen refused it, The Daily News refused it, Heinemann refused it. Thinking it might be better to find a literary agent, Lindsay was then refused by Grant Richards before being taken on by Ronald Massey. The publisher John Long accepted Sphinx in April 1923, and published it later that year.
Nicholas Cabot is working to invent a machine for recording deep-sleep dreams, but his researches are interrupted by dramatic events in the life of Lore Jensen, the flagging composer of an intriguing musical piece called "Sphinx" — events whose conclusion is being prefigured and interpreted, in a sort of symbolic drama, in Cabot's recorded dreams.
More detailed plot summary here.