"'Tommy,' I sang - it was an old song of W.B. Fair's - 'Tommy, make room for your uncle.'"
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W.B. Fair was chairman and master of ceremonies at the Royal Music Hall in High Holborn (which became the Royal Holborn Theatre of Varieties in 1887). He was also the proprietor of the Winchester Music Hall on Southwark Bridge Road between 1878 and 1880.

The song Tommy, make room for your uncle tells of a character Fred Jones, Hatter of Leicester Square, who is wooing Tommy's mother; the chorus went like this:

Tommy, make room for your uncle
There's a little dear.
Tommy, make room for your uncle
I want him to sit here.

As is implied in the text, there is a joke here because of the double meanings of the words Tommy (or Tom) and uncle.