The circulating library in Ryde

 That's Ryde, IoW, in England, not the suburb of Sydney.

I'd previously mentioned that while cataloguing a Victorian triple decker up at the Athenaeum, one volume had come from Mudie's, one from a Mudie's franchisee in Weymouth and one from a circulating library in Union Street Ryde.

Identifying the library has been tricky, but a discussion with Ann Barrett of the Ryde Historical Society has confirmed that the most likely source is either the long established circulating library at the Assembly rooms in Ryde, or another more commercial enterprise run initially by the Misses Gibbs.

Crucially, neither of them seem to have been Mudie's franchisees, suggesting that they were buying second hand books from Mudie's and perhaps elsewhere, and then selling them on for export to Australia ...