Perhaps the greatest of the Sherlock Holmes mysteries is this: that when we talk of him we invariably fall into the fantasy of his existence."Perhaps there's a reason for that, T.S.
-- T.S. Eliot, in a review of The Complete Sherlock Holmes Short Stories, 1929
Many commentators have noted T.S. Eliot's devotion to Sherlock Holmes. For example, the great poet's Old Possum's Book of Practical Cats contains a verse about "Macavity: The Mystery Cat," who is identified in the last line as "the Napoleon of Crime."
In his verse play, Murder in the Cathedral, Eliot has a long passage obviously drawn from the Musgrave Ritual. And the use of the word "Grimpen" in the poem "Little Gidding" seems to come from the Great Grimpen Mire in The Hound of the Baskervilles.
But I may have discovered a new connection between Eliot and Holmes.
In "The Adventure of the Priory School," Holmes notes that his bank is Capital and Counties on Oxford Street. Leslie S. Klinger's New Annotated Sherlock Holmes points out that Capital and Counties, also ACD's bank, was acquired by Lloyds Bank in 1918.
Young T.S. Eliot worked at Lloyds from 1917 to 1925. Presuming that Holmes continued his relationship with Capital and Counties into his retirement years, and stayed with the successor bank, T.S. Eliot could have been his banker.
Someone else may have already made this connection, but I am not aware of it.